il\ci\;ri^-i^li H «« ?.!•* "p NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1218 LECTURE SERIES «BOU]SIDARY LAYER THEORY" PART n - TURBULENT FLOWS By H. SchlLchting Translation of "Vortragsreihe" W.S. 1941/42, Luft- fahrtforschungsanstalt Hermann Goring, Braunschweig Washington April 1949 ^^SHY OF FLORIDA roajMEMTS DEPARTMENT eAINESV(LLE,FL 32611-7011 udA -EkCk TM Wo. 1218 TABLE OF CONTENTS Chapter XIII. GENERAL REMARKS ON TURBULENT FLOWS 1 a. TurlDulent Pipe Flow 1 b. TurlJulent Boundary Layers 3 Chapter XIV. OLDER THEORIES 1^ Chapter XV. MORE RECENT THEORIES; MIXING LENGTHS 11 Chapter XVI. PIPE FLOW I8 a. The Smooth Pipe I8 h. The Rough Pipe 25 Chapter XVII. THE FRICTION DRAG OF THE FLAT PLATS IN LONGITUDINAL - FLOW '. 30 a. The Smooth Pipe 3I b. The Rough Pipe J+O c. The AdmisBihle Roughness l+l Chapter XVIII. THE TURBULENT FRICTION LAYER IN ACCELERATED, RETARDED FLOW U3 Chapter XIX. FREE TURBULENCE • I+9 a. General Remarks: Estimations . . . . » h9 h. The Plane Wake Flow 56 c. The Free Jet Boundary 6I d. The Plane Jet 65 Chapter XX. DETERMINATION OF THE PROFILE DRAG- FROM THE LOSS OF MDMENTUM ■ . . 66 a. The Method of Betz 69 h. The Method of Jones 72 Chapter XXI. ORIGIN OF TURBULENCE 7I+ a. General Remarks 7^ h. The Method of Small Oscillations 76 c. Results 91 Chapter XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION LAYER ACCORDING TO THE MEl'HOD 01'" GRUSCHWITZ (R-EFERENCE 78) 93 a. Integration of the Differential Equation of the Turbulent Boundary Layer 9j b. Connection Between the Form Parameters rj and H = i3/5* of the Boundar-y Layer 96 Digitized by tlie Internet Arcliive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/lectureseriesbouOOunit MTIONAL AUVTSOEY COMMITTEE FOE AEEOKAliriCS TECinnCAL MEMOEAMUM NO. 12l8 LECTUEE SERHS "BOIMDAEY LAYER THEOEY" Part II - Turbulent Flows* By H. Schlichting CHAJTEE XIII. GEHERAL REMAEES ON TUKBIOnST FLOWS a. Turbulent Pipe Flow The flow laws of the actual flows at high Eeynolds numbers differ considerably from those of the laminar flows treated In the preceding part. These actual flows show a special characteristic, denoted as "turbulence." The character of a turbulent flow is most easily imderstood in the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. In order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity (compare chapter I, Part I) . Actually, however, one observes that, for larger Eeynolds numbers, the pressure drop Increases almost with the square of the velocity and is very much larger than that given by the Hagen— Polseuille law. One may conclude that the actual flow is very different from that of the Polseuille flow. The following test. Introduced by Eeynolds, . is very instructive: If one inserts into the flowing fluid a colored filament one can observe, for small Eeynolds numbers, that the colored filament Is maintained down- stream as a sharply defined thread. One may conclude that the fluid actually flows as required by the theory of laminar flow: a gliding along, side by side, of the adjoining layers without mutual mixing (laminar = layer flow). For large Eeynolds numbers, on the other hand, one can observe the colored filament, even at a small distance downstream from the inlet, distributed over the entire cross section, that Is, mixed *"Tortragsrelhe 'Grenzschichttheorle. ' Tell B: Turbulente Stromungen." Zentrale fiir wlssenscliaftliches Berichtswesen der Luftfahrtforschimg des Goneralluftzeugmeisters (ZWB) Berlln-rAdlershof , pp. 15^+— 279. The original language version of this report is divided Into two main parts. Tell A and Tell B, which have been translated as separate WACA Technical Memorandums, Nos. 1217 and 12l8, designated Part I and Part II, respectively. This report is a continuation of the lecture series presented in part I, the equations, figures, and tables being numbered in sequence from the first part of the report. For general information on the series, reference should be made to the preface and the Introduction of Part I. 2 NACA TM no. 12l8 to a great extent vlth tlie rest of the fluid. Thus the flow character lias changed completely for large Eeynolds numbers: A pronounced transverse mixing of adjacent layers takes place. Irregular additional velocities in the longitudinal and transverse directions are superposed on the main velocity. This state of flow is called turhulant. As a consequence of the mixing the velocity is distributed over the cross section more uniformly for turhulent than for laminar flow (compare fig. 17, part -I) . For turhulent flow there exists a very steep velocity increase in the immediate neighhorhood of the wall and almost constant velocity in the central regions. Consequently the wall shearing stress is considerably larger for turhulent than for laminar flow; the same applies to the drag. This follows also from the fact that in turbulent flow a considerable part of the energy is used up in maintaining the turbulent mixing motion. The exact analysis of a turbulent flow shows that at a point fixed in space the velocity is subjected to strong irregular fluctuations with time (fig. 72). If one measures the variation with time of a velocity component at a fixed point in space, one obtains, qualitatively, a variation as shown in figure 72. The flow is steady only on the average and may be interpreted as composed of a temporal mean value on which the irregular fluctuation velocities are superimposed. The first extensive experimental Investigations were carried out by Darcy (reference 60) in connection with the preliminary work for a large water-distributing system for the city of Paris. The first quanti- tative experiments concerning laminar pipe flow were made by Hagen (reference 95)- The first systematic tests regarding the transition from laminar to turbulent flow were made by Osborn Eeynolds (reference 6l) . He deteimlned by experiment the connection between flow volume and pressure drop for turbulent flow and investigated very thoroughly the transition of the laminar to the turbulent form of flow. He found, in tests of various velocities and in pipes of various diameters, that transition always occurred at the same value of the Eeynolds number: — . This Eeynolds number is called the critical Eeynolds number. The measurements gave for the pipe flow: Ke.^i^ = (^) = 2300 (13.1) crit v^ y /crit For Ee < Ee^^-j^^ the flow is laminar, for Ee > I^e^-^j^-j., turbulent. Later on it was ascertained tliat the numerical value of Ee^^-j^j^^ is, moreover, very dependent on the particular test conditions. If the entering flow was very free of disturbances, laminar flow could be maintained up to Ee = 24000. However, of main interest for the technical applications is tlio lowest critical Eeynolds number existing for an arbitrary disturbance of tlie entering flow, due either to irregularities in the approaching flow or to vortices forming at the pipe inlet. Concerning the drag law of the pipe Eeynolds foimd tliat the pressure drop is proportional to the 1.73 power of tlie mean flow velocity: -1-73 A"D ---. u NACA TM Wo. 1218 Id. Turtulent Boundary Layers Becently it was determined that the flow along the surface of a body (■boimdary layer flow) also can be turbulent. We had found, for instance, for the flat plate in longitudinal flow that the drag for laminar flow is proportional to \J '^q (compare ecjuation (9.I8), Part I.) However, towing tests on plates for large Reynolds numbers carried out by Froude 1 B'l (reference 96) resulted in a drag law according to which W ~ Uq ' ' . More- over, the drag coefficients in these measurements remained considerably higher than the drag coefficient of the laminar plate flow according to equation (9.I9), Part I. Presumably this deviation is caused by the turbulence of the boundary layer. A clear decision about the turbulent flow in the boundary layer was obtained by the classical experiments of Eiffel and Prandtl concerning the drag of spheres in 19114. (reference 62). These tests gave the following results regarding the drag of spheres (compare fig. 73). The curve of drag against velocity shows a sudden drop at a definite velocity V^pit.* although it rises again with further increasing velocity. If one plots the drag coefficient c^^ = w/f ^ U^ (F = frontal area) against the Reynolds number Uod/v, c^ shows a decrease to 2/5 of its original value at a definite Reynolds number (^®crit^' ^s^^L'tl explained this phenomenon in 1911j-. He was able to show that this drag decrease stems from the laminar boiindary layer changing to turbulent ahead of the separation point. The resulting considerable rearward shift of the separation point causes a reduction of the vortex region (dead water) behind the sphere (fig. 7k). This hypothesis could be confirmed by experiment: by putting a wire ring on the sphere (sphere diameter 28 centimeters, wire diameter 1 millimeter) one could attain the smaller drag at smaller V^^^^j^^ and Pe^j,^.^^. The wire ring is put on slightly ahead of the laminar separation point; it causes a vortex formation in the boundary layer, which is thus made turbulent ahead of the separation point and separates only farther toward the rear. By means of the wire ring the boundary layer is, bo— to— say, "infected" with turbulence. Due to the mixing motions which continually lead high velocity air masses from the outside to the wall, the turbulent boundary layer is able to overcome, without separation, a larger pressure increase than the laminar boundary layer. The turbulence of the friction layer is of great Importance for all flows along solid walls with pressure increase (diffuser, wing suction side). It is, however, also present in the flow along a flat plate where the pressure gradient is zero. There the flow in the boundary layer is laminar toward the front, experiencing transition to the turbulent state further downstream. Whereas the laminar boundary layer thickness increases downstream with xl/^^ the turbulent boundary layer thickness increases k WACA TM No. 1218 approximately as x ' ; that is, for the t'orb-olent bo'ondary layer the increase of the houndary layer thickness is considerably larger, (fig. 75). The position of the transition point x ., is given by (fig. 75): (13.2) In comparing the critical Eeynolds numbers for the pipe and the plate one must select r and S, respectively, as reference lengths. The equation for the flat plate is, according to Blasius (reference 8) (compare equation (9.21a)) or U^5 _ . U^x ■'o y (13.3) n Thus, with (— ) = 25000 : ^ ^ ^crit (^) = 5.500 = 2500 (flat plate) (13.^) ^ -^crit ^max ^ This critical Eeynolds number must be compared with — at the transition point for the pipe. Due to the parabolic laminar velocity distribution in the pipe ri ^ = 2 u, and because r = i d, then, for the pipe, u r/v = ud/v. According to equation (I3.I), (•^) = 23OO crit for the pipe. Thus the comparable critical Reynolds numbers for pipe and plate show rather good agreement. CHAPTER XrV. OLDER THEORIES The first efforts toward theoretical calculation of the turbulent flows go back to Reynolds. One distinguishes in the theory of turbulence two main problems: 1. The flow laws of the developed turbulent flow: The space and time velocity variations affect the time average of the velocity; they act like an additional internal friction. The problem is to calculate the local distribution of the time average of the velocity components, and thus to gain further information concerning, for instance, the friction drag. NACA TM No. I2l8 2. Origin' of turbulence: One Investigates under what conditions a small disturbance, superposed on a laminar flow, increases with time. According to whether or not the disturbance ir. "reases with time, the laminar flow is called unstable or stable. The :':vestigation in question is therefore a stability investigation, made to clarify theoretically the laminar— turbulent transi- tion. These investigations aim particularly at the theoretical calculation of the critical Eeynolds number. They are, in general, mathematically rather complicated. The first problem, since it is the more important one for general flow problems will be our main concern. The second will be discussed briefly at the end of the lecture series. As to the first problem, that of calculation of the developed turbulent flow, one may remark quite generally that a comprehensive theoretical treatment, as exists for laminar flow. Is not yet possible. The present theory of the developed turbulent flow must be denoted as semi— empirical. It obtains its foundations to a great extent from experiment, works largely with the laws of mechanical similarity, and always contains several or at least one empirical constant. Nevertheless the theory has contributed much toward correlating the voluminous experi- mental data and also has yielded more than one new concept. For the numerical treatment one divides the turbulent flow, unsteady in space and time, into mean values and fluctuation quantities. The mean value may a priori be formed with respect to either space or time. We prefer, however, the time average at a fixed point in space, and form such mean values of the velocity, pressure, shearing stress, etc. In forming the mean values one must not neglect to take them over a suffi- ciently long time interval T so that the mean value will be independent of T. Let the velocity vector with its three mutually perpendicular components be •»v = iy + Jv + kw (1^-1) For a tijrbulent flow the velocity components are therefore functions of the three space coordinates and the time: u = u (x, y, z, t) V = V (x, y, z, t) (li+.2) w = w (x, y, z, t) The time average for the component u, for instance, is formed as follows: (li^.3) *nTroughout the text, underscored letters are used in place of corresponding German script letters used in the original text. WACA TM No. 1218 If u, V, w are independent of t^ and T, the motion is called steady on the average J or quasisteady. A steady turbulent flow^ in the sense that the velocity at a point fixed in space is perfectly constant, does not exist. The velocity fluctuations are then defined by the equations u = u + u' T = V + V w = w + w (ih.k) and in the same way for the pressure: P + P' (1^.5) The time average of the fluctuation quantities equals zero, according to definition, as the following consideration will show immediately: to+T ,to+T u'dt = - Ito+T udt-^ dt = u - u = {Ik. 6) ThuE u- (ik.V The Additional "Apparent" Turbulent Stresses As a result of the velocity fluctuations additional stresses ( = apparent friction) originate in the turb'olent flow. This is readily illustrated for instance by the case of the simple shearing flow u = u(y) (fig. 76). Here ~ = 0; however, a fluctuation velocity v' in the transverse direction is present. The latter causes a momentum transfer between the adjoining layers across the main flow. This momentum transfer acts like an additional shearing stress t. Whereas in laminar flow the friction is brought about by the molecular momentum exchange, the turbulent exchange of momentum is a macroscopic motion of, mostly, much stronger effect. The equations of motion of ths turbulent flow, with this tiirbulent appa.rent friction taken Into consideration, can be obtained from the Navler— Stokes differential equations by substituting equation (lli.lj-) into the latter and then forming the timc^ averages in the Wavier— Stokes ilfferentlal equations. To that purpose the Navier— Stokes differential equations (3.I6) are written in the form: NACA TM No. 12l8 'Sa ^(u^) S(uv) S(uw) p|^ " ~ax az P T— + — ^ + + — ^ - [at Bx Sy hz St Bx Sy Sz ap /S^u B^u sV ^ + ^^ ( -2 + -^ + — 2 \Sx Sy Sz / Sp /S2t B2v S2v^ — + n' — - + + \Sx Sy Sz > \Sx Sy Sz -— + — + — = ox Sy Bz > (1^.8) By introducing equations (l^A) and (lli-.5) and forming the time averages one first ottains from the continuity equation: 5u Sv c)w _ Sx Sy Sz (11..9) and thus also: Su' St' Sw' + + = Sx Sy (11^.10)- By introduction of equations (lii.U) into the left side of equation (14.8) one obtains expressions asj for instance, 2 2 2 — 2 u = (u + u' ) = u + 2 u u' + u' etc. In the subsequent formation of the time average the squared terras in the "barred quantities remain unchanged since they are already constant with respect to time. The mixed terms^ as for Instance u u', . . . and also the terms that are linear in the fluctuation quantities are eliminated in forming the average "because of equation (1^.7). However, 2 the terms that are quadratic in the fluctuation quantities as u' , a*v*, . . . remain. Thus one o"btains from the equation system (equation (ll4-.8))j after forming the time average the following system of equations: 8 KACA TM No. 1218 V dx 'dj 3z /_5t -Bv — Sv Sx 5y Sz — + uAu — p < Sx hj Sz ^ + MAT -p p^^ + ^ll- + ^'^' Sy Sx ^ Sz /_^ _^ -5w 1 Bp .- plu^- + V-— + w— 1= - -ii + pAw -p \ ox oj oz / oz 2 I Sx Sy Sz ^ > (14.11) The left side now formally agrees with the Navier-Stokes differential equations for steady flow if one writes instead of u, y, w the time averages of these quantities. On the right side additional terms which arise from the fluctuations have been added to the pressure and friction terms . Remembering that in deriving the Wavier— Stokes differential equations one could write the resultant surface force per unit volume "by means of the components of a stress tensor according to equation (3.7) in the form 5x Sy St Sa Bt ,jl^Z^ y. yz Sx k \ + — ^— + Sy Sz -^ Bx ^ Bz> > ilk.l2) one recognizes "by comparison with (equation (lU.ll)) that one may introduce for the quantities added by the fluctuation motion a syiimietrical stress tensor in the following manner: ,2 a = — pa' X ^ T = — pu'v' xy V " " ^'^'^' ay = - pv'2 T = — pu'w' XZ ^ yz pv^^'ir tTJ« T = — pu'w' XZ Ty^ = - PV'W' a^ = - pw»2 ^ J (14.13) One has therefore^, for the mean values of the quaslsteady flow, the following equations of motion: NACA TM No. 12l8 , -Su -Su -Su PJU— + T— + W— 3X aj oz /-Civ' _Sv -Sv P(U + T + W Sx Sy Sz Ai^ + v^ + w^V Bx Sy Sp ^- ^^x ox dx St Bt xy Sy St Sp A— " xy By Sx By Bp + ^v + -^ + -J- Bz Bx By JLZ xz Bz Bt + yz Ba. z Bz > (1^.1^) y The continuity equation (equation (li<.,9)) also enters. The boundary conditions are the same as for laminar flow: adhering of the fluid to the wall, that is, on the solid walls all velocity components equal zero. According to equation (l^.lif) the mean valuep of the turbulent flow obey the same equations of motion as the velocity component's of a laminar flow, with the friction forces, however, increased by the apparent stresses of the turbulent fluctuation motion. But since the fluctuation velocities u*, v', . . . and particularly their space distribution are unknown, equations (lij-.lij-) and (l4.13) are, at first, rather useless for tlie calculation of a turbulent flow. Only when one will have succeeded in expressing the fluctuation quantities "' u' ~, u'v', ... in a suitable manner by the time averages will it be possible to use equations (l4.lU) to calculate. in particular, the mean values u, v, w. A first expression of this irind which brought however little saccess was originated by Boussinesq (reference 6k). He introduced, aside from the ordinary viscosity coefficient, a new viscosity coefficient of the apparent turbulent friction. In analogy to the stress tensor for laminar flow which is, according to equation (3.I3): cr T T X 'xy xz ''xy ^y ""yz ""xz "^jz ^z p — — p p + u- Bu Bu Bu Bx By Bz Bv Bv Bv Bx By Bz Bw Bw Bw Bx ^y Bz + 1^ Bu Bv Bw Bx Bx Bx Bu By Bv Bw By By Bu Bz Bv Bz Bw Bz ^^^ (14.15) Boussinesq puts for the apparent turbulent friction: Or. Bu ^ _ „ / Bu , Bv X dx xy \$,j 3z (1I+.16) Then there corresponds to the laminar viscosity coefficient n the mixing factor pg; 10 NACA TM No. 12l8 H~pe or v~e The kinematic viscosity of the turhulent flow (apparent friction) is usually very much larger than that for the ordinary laminar friction. (Hundred— or thounsandfold or more). In general, one may therefore altogether neglect the ordinary viscosity terms mAu, ... in equation (lij-.li<-). Only at the solid walls where due to the no— slip condition u=v=w=0 as well as u* = v' = w' =0 the apparent turbulent friction disappears, does the laminar friction again hecome dominant. Thus there exists in every turbulent friction layer in the immediate neighborhood of the wall a very narrow zone where the flow is laminar. The thickness of this laminar sublayer is only a small fraction of the turbulent boundary layer thickness. One can easily understand from the example of the simple shear flow according to figure 76 that in a turbulent flow the mean value u'v» is different from zero. For this case, a correlation exists between the fluctuation velocities u' and v' in the following manner: The particles with negative v' have "mostly" a positive u*, since they come from a region of larger mean velocity u.* The parts with positive v', on the other hand, have "mostly" a negative u', because they come from a region with smaller u, and retain in the transverse motion approximately the x-momentum of the layer from which t hey come. Thus, "mostly" u'v' < and, therefore, the time average u'v* < 0. Therefore, the shearing stress is this flow is: = - p u'v' > xy In measuring turbulent flows one usually measiH'es only the mean values u, V, . . . since only they are of practical interest. However, in order to obtain deeper insight into the mechanism of the turbulent flow, the fluctuation quantities have recently been measured and also their mean squares and products: V^.2 v^ v^ According to measurements by Eelchardt (reference 65) in a rectangular tunnel (width 1 millimeter, height 2k centimeters) the maximum value ^"Mostly" is to indicate that particles with different signs, thoui^Ji not excluded, are in the minority. NACA TM No. 12l8 11 \[^'- of V li* J for instance, equals 0.1 3 vl^, the majcimum -value of equals 0.05 u^^^. Both naxima lie in the neighborhood of the wall. One may saj, therefore, that in this case the turhulence is strongest near the wall. In a flow that is homogeneous (wind t'onnel), turbulent fluctuations are also always present to a varying degree. They determine the so— called degree of turbulence of a wind tunnel. Since the measurement of the fluctuation quantities is rather difficult (hot— wire method), a more ■ convenient measuring method has been chosen, for the present, for determin- ation of the degree of turbulence of a wind tunnel: namely, the determin- ation of the critical Eeynolds number for the sphere from force measure- ments or pressure distribution measurements. One defines as critical Reynolds number the one where the drag coefficient c^ = 0.3. It becomes clear that a unique connection exists between the critical Reynolds number and the turbulent fluctuation velocity in the sense that the critical Reynolds number of the sphere is the lower, the higher the turbulent fluctuation velocity. According to American measorements (reference 97) the connection between the longitudinal fluctuation and the measured critical Reynolds number for the sphere is as shown in the following table: i.'/s O.OOij- 0.0075 0.012 0.017 0.026 «%rlt ^°"^ 2.8 2.k 2.0 1.6 1.2 In addition to the apparent Increase of viscosity, the turbulent fluctuation motion has other effects: It tends to even out any tempera- ture differences or variations in concentration existing in a flow. The diffusion of heat, for instance, is much larger than for laminar flow, because of the exchange motions which are much stronger in turbulent flow. A close connnection therefore exists, for Instance, between the laws of flow and of heat transfer from a heated body to the fluid flowing ■by. Ninth Lecture (February 2, 19i<-2) CHAPTER XV". MORE RECENT THEORIES j MIXING LENGTH In order to make possible a quantitative calculation of turbulent flows, it is necessary to transform the expressions for the apparent turbulent stresses (equation (ll4-,13)) in such a manner that they no longer contain the unknown fluctuation velocities but contain the components of the mean velocities. Consider, for that purpose, a particularly simple 12 MCA TM No. 12l8 flow, namely a plane flow which has the same direction everywhere and a ■velocity varying only on the different stream, lines. The main— flow direction coincides with the z— direction; then: u = il(y) V = w = (15.1) Of the shearing stresses, only the component t = t" ia present, for which from equation (114-.13) as well as from Boussinesq's equation, equation (llf. 15). there resiilts: du T = - p u'v» = p e ii^ (15.2) dy This formula shows that | t |/p equals the square of a velocity. One putSj therefore, for use in later calculations. u»v' (15.3) and denotes v.,,. as shearing stress velocity. Thus this shearing stress velocity is a measure of the momentum transfer "by the turhulent fluctuation motion. According to Prandtl (reference 66), one may picture the turhulent flow mechanism, particularly turhulent mixing, in the following simplified manner: Fluid particles, each possessing a particular motion, originate in the turbulent flow; they move for a certain distance as coherent masses maintaining their velocity (momentum). One now ass-jmes that such a fluid particle which originates in the layer (y, — l) and has the velocity u(y-| — l) moves a distance I = mixing length normal to the flow (fig. 77)' If this fluid particle maintains its original velocity In the x— direction it will have, in its new location y^, a smaller velocity than Its new surroundings, the velocity difference heing -u' = uij^) - u(y-^ - I) with v' > Likewise a fluid particle coming from the layer (y-, + l) to y-|^ has at the new location a greater velocity than the surroundings there; the difference Is NACA TM No. 12l8 13 Tij and u^ give the turbulent velocity fluctuation in the layer y, . One ohtalns for the mean value of this velocity fluctuation |u'| = i (|u^|+ lu^l) = I (t)l (I5.i^) From thla equation one ohtains the following physical interpretation for the mixing length Z: The mixing length signifies the distance in the transverse direction which a fluid particle must travel at the mean velocity of its original layer so that the difference "between its velocity and the velocity of the new location equals the mean velocity fluctuation of the turhulent flow. It is left open whether the fluid particles in their transverse motion fully maintain the velocity of their original layer, or whether they have partly assumed the velocity of the traverse layer and then travelled larger distances in the transverse direction. The Prandtl mixing length which is thereby introduced has a certain analogy to the mean free path of the kinetic theory of gases, with, however, the difference that there one deals with microscopic motions of the molecules, here with macroscopic motions of larger fluid particles. One may picture the origin of the transverse fluctuation velocity v' in the following way: Two fluid particles flowing from the layers (y-, + l) and (y-, — Z) meet in the layer y-[_ in such a manner thab one lies behind the other: the faster (y-|_ + Z) behind the slower (y-i — 2). They then collide with the velocity 2u' and glveway laterally. Thereby originates the transverse velocity V , directed away from the layer y-, to both sides. If, conversely, the slower of the two particles is behind the faster, they withdraw from each other with the velocity 2u' . In this case the space formed between them is filled up out of the surroundings. Thus originates a transverse velocity v* directed toward the layer y-,. One concludes from this consideration that v' and u' are of the same order of magnitude and puts |v'| = number |uM = ntunber Z — (I5.5) In order to express the shearin g str ess according to equation (I5.2) one has to consider the mean value u*v* more closely. The following conclusions can be drawn from the previous considerations. Ik NACA TM No. 12l8 The particles arrlTing in the layer y-, with positive t* (from "below, fig. 77) have "mostly" a negative u' so that u'v' is negative. For the particles arriving with negative v'j u* is "mostly" positive, so that u'v' is again negative. "Mostly" signifies that particles with different sign are not wholly excluded, but are strongly outnumbered. The mean value u'v» is therefore different from zero and negative. Thus one puts u»v« = - k lu'l Iv'l (15.6) with k^O; 0<k<l. The niimerical factor k, also called the correlation coefficient is not known more closely. According to equation (I5.5) ^^^ (15»^) one now obtains u«v' = r,\2 - number Z^/duV (15.7) the "number" in this equation being different from the one in equation (15.5)* If on.e includes the "number" in the unknown mixing length, one can also write ^TTTT = _ 22/duJ (15.8) and thus finally obtains for the turbulent shearing stress according to equation (I5.2) P 2 2/du\2 Considering that the sign of t also must change with the sign of it is more correct to write du T=pi2 du iy du Prandtl's ^7 formula (15.9) This Is the famous Prandtl mixing length formula which has been very successful for the calculation of turbulent flows. If one compares this formula (equation (15.9)) with the equations of Boussinesq where one had put t = e -r— ( e = mixing factor = turbulent analogue of the laminar viscosity \x) , one has for the mixing factor RACA TM Ro« 12l8 15 (15.10) c = p2^ du The turlaulent mixing factor e Is in moat cases larger than the laminar ■viscosity [x "by several powers of ten. Moreover the mixing factor e is dependent on the velocity and on the location and tends toward zero near a wallj "because there the mixing length goes toward zero. If one compares Prandtl's formula equation (I5.9) with Boussinesq«s equation (15.2) one could f)erhap8 think at first that not much has been gainedj since the unknown quantity e ( = apparent viscosity) has been replaced by the new unknown I = mixing length. Nevertheless Prandtl's formula is considerably better than the old formula for the following reason: It is known from tests that the drag for turbulent flow is proportional to the square of the velocity. According to equation (15.9) one obtains this square law for drag by assuming the mixing length to be independent of the velocity, that is, by assuming the mixing length to be purely a function of position. It is considerably easier to make a plausible assumption for the length I = mixing length than for the apparent turbulent viscosity e, and therein lies the considerable superiority of Prandtl's formula equation (I5.9) over Boussinesq's equation (I5.2). In many cases the length 2 can be brought into a simple relation to the characteristic lengths of the respective flows. For the flow along a smooth wall I must, at the wall itself, equal zero, since all trans- verse motions are prevented at the wall. For the flow along a rough wall, however, the limiting value of I at the wall equals a length of the order of magnitude of the height of the roughness. It would be very useful to have a formula permitting the determina- tion of the dependence of the mixing length on the position for any arbitrary flow. Such an attempt has been made by v. Karman (reference 68). V. Karman makes the assumption that the inner mechanism of the turbulent flow is such that the motion at various points differs only with respect to time— and length-scale, but is otherwise similar (similarity hypothesis). Instead of the units of time and length one may select those of velocity and length. The velocity unit that is Important for the turbulent motion is the shearing stress velocity v^ according to equation (I5.3). The corresponding unit of length is the mixing length Z. In order to find the quantity I from the data of the basic flow u(y), V, Karman applies the Taylor development* for u(y) in the neigh- borhood of the point y-, . u(y) = u(y^) + (y - y^) ^0) ^ir('^- ^1^^ (^) + ^15.11) * In the following, the bar over the mean velocity will be omitted, for simplification. 16 NACA TM No. 12l8 The length 2 cannot depend on the velocity ^.(y-,), since according to Newton's principle of relativity the addition of a constant velocity has no influence on the course of motion. Thus ■du\ and the higher derivatives remain as characteristic data of the "basic flow. The simplest length to "be formed from it is du dy d!u dy^ V. Earman puts therefore (15.12) According to this formula I is not dependent on the amount of velocity "but only on the velocity dlstrl"butlon. Thus 2 is a pure position function as required ahove. In equation (15.I2) k, is an empirical constant which must "be determined from the experiment. To arrive further at the tur"bulent shearing stress, v. Kannan also maintains Prandtl's equation (15.9). In generalizing equation (I5.9) one o"btains, according to Prandtl, the complete expression for the tur"bulent stress tensor of a plane flow in the form /CT^ T, \:^ % \= pi' du dy Sx Su I ^ Si ^ (15.13) du Sv ^ dx '" Sy 2 ^ The common factor on the right side signifies the tur"bulent mixing factor according to equation (15.IO). NACA TM No. 12l8 I7 Tenth Le6ture (February 9, 19k2) Flow Along a Smooth Wall ¥e will immediately make a first application of Prandtl's formula (equation (15.9)) for the flow along a smooth wall. The normal distance from the wall is denoted as y. Let the wall coincide with the x-axis. For the Telocity distribution then, u = u(y). For this case one sets the mixing length in the neighborhood of the wall proportional to the distance from the wall I = K.J ' (15.1^) the constant k must he determined from the experiment. Moreover one makes the assumption that the shearing stresb t is constant in the entire flow region; then the shearing stress Telocity v^ according to equation (15.3) also is constant. If one further neglects the laminar friction^ one obtains from equations (I5.2), . (I5.9), (15.14) or ^ = I* dy Kj and by integration u = — 2n y + constant (I5.I5) In determining the constant of integration one must pay attention to the fact that the turbulent law equation (I5.9) does not apply right up to the wall but that Tery near to the wall an extremely thin laminar layer is present. From the laminar Tlscoslty \x and the turbulent shearing stress Telocity t^ one can form the length v/t^. The constant of Integration in equation (I5.I5) is determined from the condition that u = for y = yQ. Thus there results, according to equation (I5.I5) u = ^ (2n y - Zn y^^) (15-16) The as yet unknown distance from the wall yQ is set proportional to the length v/t^, thus 18 NACA TM No. 12l8 yo = P =fe - (15.17) where 3 signifies a dimensionless constant. Thus one finally ottains for the velocitj disbritution at the smooth wall T^ y^* u = ^ (in ±^ - in ^) (15.18) that is, a logarithmic Telocity distribution law. It contains two empirical constants k. and p. According to measurements «; = O.k. From equation (I5.I8) one can see that the dimensionless velocity u/v>. = cp can he represented as a function of the dimensionless distance from the wall T\ = v^(.y/v. The latter is a sort of Eeynolds number, formed with the distance from the wall y and the shearing stress velocity v^.. Thus one obtains for larger Reynolds numbers from equation (I5.I8) the following universal velocity distribution law Cp(Ti) = A In T\ + B (15.19) with A = l/n = 2.5. For smaller Eeynolds numbers, where the laminar friction also has a certain influence, tests gave the velocity distribution larf Cp(Tl) = C Tl'' (15.20) or t - ^ (^Y (15.20a, with the exponent n equalling about l/?. These universal velocity dis- tribution laws according to measurements for pipe flow are given in figure 78. They will be discussed in more detail in the following chapter. CHAPTER rVI. PIPE FLOW a. The Smooth Pipe Among the various turbulent flows of practical importance, pipe flow was investigated with particular thoroughness because of its great practical importance. We shall therefore consider the pipe flow first. It will be noted at this point that the flow laws of the pipe flow may be applied to other cases, as for instance the plane plate in longituainal flow. Consider a straight pipe of circular cross section and with a smooth wall. Let y be the radial coordinate measured from the pipe axis. The WACA TM No. 1218 19 "balance of forces "between the shearing stress t and the pressiire drop P-, — Pp on a piece of pipe of length L yields as "before for the laminar flow according to equation (2..1a)j the relation: -^-^1 (16.1) This formula applies equally to laminar and turbulent flow. In it t now signifies the sum of the laminar shearing stresses and of the apparent turhulent shearing stresses. Over a cross section, t is proportional to y. The shearing stress at the wall Tq may "be determined experimentally "by measurement of the pressure drop: Pn - P2 r r^=-^—^L . (16.2) For the tur'bulent flow the connection "between pressure drop and flow volume Q = irr u must "be obtained from tests.* In the literature there exists a very great number of pipe resistance formulas. Only those serve our purpose which satisfy Eeynolds' law of similarity. One of them is the formula of H. Blasius (reference 69), set up partic-ularly carefully, which is valid for a smooth wall and for Eeynolds numbers Ee = ud/v^ 100 000. If one introduces, as "before in equation (2.6), the dimensionless pipe resistance coefficient X "by the equation !Ll^ = iPu2 (16.3) L d 2 . X is, according to Blasius: ,- ,x-iA X = 0.316U(^^J (l6.k) Comparing equations (l6.2) and (I6.3) one finds: ^o = I P ^^ ■ (16-5) and therefore according to equation (iG.k): _7/k lA -lA Tq = 0.03955 p a'^ V ' d (16.6) *In the following, u is, for the pipe flow, the mean flow velocity at the cross section, as distinguished from the time average in the previous sections. 20 NACA TM No. 12l8 If one introducsBj moreover, instead of the diameter d the radius r, the nximerical factor in this linear equation must be divided by 1/4 2 ' =1.19. Thus T becomes: o ^o = 0.03325 P u^/^ V^A r-lA = p v^2 (ig^^) where the shearing stress velocity is defined by the wall shearing stress: v^ = ^ (16.8) If one finally factors the quantity v^,. in equation (16.7) into v^^. ' x v^ ' , one obtains: This equation is very similar to equation (15.20a); however, the mean velocity now takes the place of the local velocity and the pipe radius takes the place of the distance from the wall. One passes first from the mean velocity to the maximum velocity u-|_; based on measurements of Wikuradse (reference 70) u = 0.8 u,, and therewith follows from equation (16.9): u, f^*r\^/^ If this formula is assumed to be valid for any distance from the wall, one obtains: ^^ , B.lk (^-ff (16.10) or <p = 8.1k y^l'^ (16.11) This is the so— called 1/7— power law for the velocity distribution; its form was already given in equation (15.20a). The coefficients n and t]. NACA TM No. 12l8 21 still uiLlaaown there^ have now "been determined on the tasis of the resistance law of the pipe flow. Figure 78 shows, according to measure- ments of Nikuradse (reference 70) that this law is well satisfied in the range of Reynolds numhers up to 100,000. Naturally this law of velocity distribution can apply only to the region of Reynolds n^jimbers where the pipe resistance law given by equation {l6.h) is valid, since it was derived from this law. For pirrposes of later calculations we shall derive from equation (16.IO) the shearing stress velocity v^. One obtains: 7/8. V 1/8 v^ = 0.150 u (^] (16.12) 7 /fi 1 with 8.7^ = 6.65 and = O.I5O. From equation (16.I2) follows: 6.65 Tq = p rf = 0.0225Pu'^/^ (tT^^ (I6.I3) \y/ This relation will be needed later. Comparing measured velocity distributions with equation (16.IO) one can state that outside of the range of validity of equation (16.IO), namely for Re > 100,000, a better approximation is obtained by the power 1/8, l/9j or l/lO instead of l/7- The measurements concerning the pipe resistance (fig. 8I) show an upward deviation from the formula of Blasius for larger Reynolds numbers. The logarithmic velocity distribution law, equation (15.19)^ derived in the previous chapter has been verified by Nilniradse (reference 70) on the basis of his meas'jrements for the smooth pipe. For this purpose from the measured press'ore drop for each velocity profile one first determines the wall shearing stress according to equation (l6.2) and from tha t, according to equation (16.8), the shearing stress velocity v^ = \/t /p. Then the dimensionless velocity cp = u/v^ can be plotted against the dimensionless distance from the wall t] = yv^(./v. The measurements of Nikuradse in a very large range of Reynolds numbers. He = k X 10^ up to 32^4-0 X 103, lie very accurately on a straight line if one plots cp against log r] (fig. 78). The straight line has the equation: cp = 2.5InTi + 5.5 (l6.11f) This gives, by comparison with equation (I5.I8), the following nimerical values for the coefficients K and P 22 NACA TM No. 12l8 K = oAoo p = 0.111 (16.15) Mixing Length From the measured Telocity distribution and the measirred pressure drop the distribution of the mixing length over the pipe cross section can he determined according to equations (l6.2)j (I6.I), and (15.9). T = Tq — (y = distance from the pipe axis). This determination of the mixing length from the meaS'jrements in the pipe was made hy Wikuradse (reference 70). For large Eeynolds numbers^ where the influence of Tiscosity is negligihle, one ohtains a distribution of the mixing length l/r over y/r which is independent of the Ee-number (fig. 79). The following interpolation formula can be given for this distrihution: - = 0,11+ _ 0.08 f 1 - ^ ) - 0.06(1 - ^) (16.16) In this equation y signifies the distance from the wall. The develop- ment of equation (16.16) for small y/r (neighborhood of the wall) gives 2 I = o.kj - o.k-k — + . . . (16.16a) In the neighhorhood of the wall the mixing length is, therefore, propor- tional to the distance from the wall. Equation (16.I6) for the distri— hution of the mixing length applies not only to the smooth pipe, hut, according to the measurements of Nikuradse (reference 71) also to the rough pipe, as can he seen from figure 79- From this fact one can derive in a very simple manner a universal form for the law of velocity distri- hution, valid for the smooth as well as for the rough pipe. One puts for the mixing length distrihution: I = Kj f('^')with f ^^^-^ 1 for ^ -^' 0. Furthermore follows from the linear distrihution of the shearing stress over the cross section: T = T^ (1 — ^j (y= distance from the wall) together with equation (15-9) <iy I Up ^ ^ (16.17) " 7f(y/r) and hence by integration: MCA TM Wo. 1218 23 u = ^ J! 1_1 (16.18) the lower limit of Intergration Jq where the velocity equals zero is, according to the considerations of the previous section, proportional to v/v^j thus: Jq/t = F-|_f— \ From equation (16.18) follows: r \ r and therefo'rej from equations (I6.I8) and (l6.l8a) %ax - u = ^* F Cy/r) (16.19) This law, with the same fimctlon F(y/r), applies equally to smooth and rough pipes. It states that the curves of the velocity distrihution over the pipe cross section for all Eeynolds numbers and all roughnesses can "be made congruent "by shifting along the velocity axis, if one plots (u^q y — u/v^ against y/r (fig. 80). The explicit expression for the function F(y/r) is ohtainsd immediately from equation (l6.1i<-)j according to which ■ u - u = 2.5v. to ^ = 5.75 -^^ log - (16.20) max jy Universal Resistance Law According to their derivation the velocity— distribution— law (equations (16.I9) and (l6.20)) are to he regarded as valid for arbitrary Eeynolds number since the laminar viscosity was neglected as compared with the turbulent viscosity. We shall now derive from the velocity- distribution— law equation (l6.20) a resistance law which, in contrast to Blasius*, applies up to Reynolds numbers of arbitrary magnitude. 2k WACA TM No. 12l8 From equation (l6.20) one may determine "by integration over the cross section the mean flow velocity u. One finds: ^ = "majc - 3.75 v» (l6.21) The test results of Nikuradse (reference 70) gave a number slightly different from 3-75^ namely: According to equation (16.5): X. = 8 (^) ^ (16.23) From the uniyersal Telocity distribution law of the smooth pipe equation (l6.l4) follows: %ax = ^* {2-5 in ~ -^ 5.5 and hence the connection with equation (16. 2l): u = T^ J2.5 In ^ + 1.75}* (16.24) The Reynolds numher enters into the calculation by msans of the identity; III ^ 1 ud T* ^ ud \[X V - 2 V ^ V i^^ Thus results from equations (16.23) and (l6.2k): 8 • X = 2.5 Zn (^^ \lxj - 2.5 In 1^ \|2 + 1.75 (2.035 log(^ \fx)-0.9l| WACA TM No. 1218 25 or: -^ = 2.035 log (^ ^) - 0.91 (16.25) Accordingly a straight line must result for the resistance law of the smooth pipe, if one plots l/ \/7, against logf^>/Xj. This is Tery well confirmed hy Nikuradse's measurement (fig. 8I). The numerical values according to the measurements differ only slightly from those of this theoretical derivation. From Nikuradse's measurements was found: (16.26) Universal Resistance La v For Smooth Pipes This is the final resistance law for smooth pipes. On the basis of its derivation it may he extrapolated up to Reynolds numbers of arbitrary magnitude. Thus measurements for larger Reynolds numbers than those of Nikuradse's tests are not required. Up to Re = 100,000 this 'juiiversal resistance law Is In good agreement with the Blasius law according to equation (16.U). For higher Reynolds numbers the Blasius law deviates considerably from the measurements (fig. 8I). Concerning the determination of X from equation (16.26), where it appears on both sides, it should be added that it can be easily obtained by successive approximation. Eleventh Lecture (February 16, 19i4-2) b. The Rough Pipe The ch.aracteristic parameter for the flow along a rough wall is the ratio of grain size k of the roughness to the boundary layer thickness, particularly to the thickness of the laminar sublayer 5 which is always present within the turbulent friction layer in the immediate neighborhood of the wall. The thickness of the laminar sublayer is 5, = number — . The effectiveness of roughness of a certain grain size depends, therefore, on the dimensionlese roughness coefficient k/6,«, In the experimental investigations of the resistance of turbulent flows over rough walls_, the rough pipe has been studied very thoroughly since it is of great practical importaiice. Besides depending on the Reynolds number, the resistance of a rough pipe is a function of the relative roughness r/k. One distinguishes for the resistance law of a rough pipe three regions. 26 NACA TM No. 12l8 The subsequently given "boundaries of these regions are Talid for sand roughness kg like those investigated "by Nikuradse (reference 71). 1. Hydraulically smooth: The grain size of the roughness is so small that all roughnesses lie within the laminar aulilayer. In this case the roughness has no drag increasing effect. This case exists for small Eeynolds numbers and for values of the characteristic roughness numher: 2. — 5. 2. Fully developed roughness flow: The grain size of the roughness is so large that all roughnesses project from the laminer auh— layer. The friction drag then consists predominantly of the form drag of the single roughness elements. A purely square drag law applies. For the pipe the resistance coefficient X, is then independent of Re and only dependent on the relative roughness k/r. This law exists for very large Resmolds numbers. For sand roughness this law applies for: — — §■ ^ 70. 3. Intermediate region: Only a fraction of the roughness elements project from the laminar suhlayer. The drag coefficient depends on r/k as well as on Re. This law exists for medium Reynolds numters andj for ^ v^ ^ the sand roughness, for: 5 — 70. The dependence of the pipe resistance coefficient on the Reynolds number and on the relative roughneBS according to the measurements of Nikuradse (reference 71) can he seen from figure 82 as well as, in particular, the three laws Just given. The velocity distrihution on a rough wall is given, hasically, in the same way as for the smooth wall hy equation (I5.I6). One has only to substitute for the constant of integration y^ another value: yQ proportional to the roughness grain size. One puts for sand roughness yQ = 7 kg and hence obtains from equation (I5.I6) (16.27) f ■V u 1 1 = — ^ 7,n J - 7,n y ? V* K ^ ^a ) The constant 7 is, moreover, a function of the roughness form and the roughness distribution. Comparison with experiments of Nikuradse (reference 71) on pipes roughened artificially by sand yields for the velocity distribution the general formula: 2.5 In ^ + B ] (16.28) MCA TM No. 1218 27 the constant B teing different in each domain descrlhed above; it depends on v^k /v . For the fully developed roughness flow B = 8.5, thus: u = v^ ^2.5 to ^ + 8.5") (fully rough) (I6.29) whence follows: %ax = ^*(2-5 ^n^+ 8.5) (16.30) and: u - u = v^ 2.5 2n - (16.30a) in agreement with equation (l6.20). Thus there applies also to rough pipes, as equation (l6.21) did hefore to the smooth pipe, the relation: '^ = %ax - 3-75 ^* (16.31) From here one can, by a calculation which is perfectly analogous to the previous one for the smooth pipe, easily arrive at the resistance law of the rough pipe for fully developed roughness flow. By insertion of u^^^ according to equation (16.3O) into equation (I6.3I) one obtains: or: or; u = v^ ("2.5 2n ^ + 1^.75) (16.32) /v \2 8 x = Q[^) =, 2 (16.33) (2.5 in ^.^.75) X = -, Tp (16. 3i^) (2.0 log — + 1.68j 28 NACA TM No. 12l8 This is the square resistance law of the fully deTeloped roughness flow. Comparison with the test results of Nikuradse (fig. 83) shows that one ohtains "better agreement if one changes the number 1.68 to 1.7i^. Thus the resistance law of the pipe flow for fully developed roughness is: (16.35) X 1 (^ log V ^s + 1 -f In the plots of -j= against log r/kg, (fig. 83) the test results fall very aGcxirately on a straight line. For flow along a rough wall in the intermediate region the constant B in equation (16.28) is, moreover, a function of the roughness coeffi- cient T^ /v . For this case also one can derive the resistancfe law immediately from the velocity distribution. According to equation (16.28); B = — - 2.5 2n -i = u ^2£ _ 2.5 to ^ * 8 (16.36) On the other hand, according to equations (16.3I) and (16.23) u max ^» ^* - + 3.75 = ^+ 3.75 (16.37) so that one ohtains from equation (I6.36) B V u o I. -,_ y_ ^ 2V2 _ o c -,„ r = ^^ - 2.5 In V-. k 2.5 2n i^ + 3.75 (16.38) ^s One can, therefore, determine the constant B as a function of v^k^S' either from the velocity distribution or from the resistance law. The plot in figure 8k- shows good agreement hetween the values determined 'dj these two methods. At the same time the determination of the resistance law from the velocity distrlhution is confirmed. The formula for B includes the case of the smooth pipe. B is, according to equation (l6.1i4-). NACA TM Wo. 1218 29 u T "^^^s B = — - 2.5 Zn f- = 2.5 In + 5.5 (16.39) Thus a straight line results for B in the plot against log t^ /v. Other Eoughnesses Because of the great practical importance of the roughness— pr oh lem a few data concerning roughnesses other than the special sand roughness will he given. Fikuradse's sand roughness may also he characterized hy the fact that the roughness density was at its maximum value^ because the wall was covered with sand as densely as possible. For many practical roughnesses the roughness density is considerably smaller. In such cases the drag then depends, for one thing, on form and height of the roughness, and, moreover, on the roughness density. It -is useful to classify any arbitrary roughness in the scale of a standard roughness. Nikuradse's sand roughness suggests itself as roughness reference (roughness scale) because it was investigated for a very large range of Reynolds numbers and relative roughnesses. Classification with respect to the roughness scale is simplest for the region of fully developed roughness. According to what was said previously, for this region the velocity distribution is given by: ^ = 5.75 ZOg ^ + Bg Bg = 8.5 (16.40) and the resistance coefficient by: r \2 2.0 log — + 1.1k j (16. Ul) One now relates to an arbitrary roughness k an equivalent sand roughness kg by the ratio kg = a k (16.42) Where by equivalent sand roughness kg is meant that grain size of sand roughness which has, according to equation (l6.4l) the same resistance as the given ro-oghness k. Basically, of course, the equivalent sand roughness kg can be determined by a reslstajice meas'jrement on the pipe. However, such measurements 30 NACA TM No. 12l8 for arbitrary roughneases are difficult to perform. Measijrements on arl)itrary roughnesses In a tunnel with plane walls are more convenient. To this purpose an exchangealile wall of a tunnel with rectangular cross section Is provided with the roughness to he Investigated (fig. 85). From the measurement of the velocity distrihution in sach a tunnel with a rough and a smooth longitudinal wall one ohtalns, for the logarithmic plot against the distance from the wall, a triangiolar velocity distrihu— tion (compare fig. 85). From the logarithmic plot of the velocity dlstrl- hutlon over the rough wall u = n^ log y + m^ (I6.U3) one ohtalns "by comparison with the universal law according to equation (I6.28) for the shearing stress velocity at the wall: *"" 5.75 Further, one determines for the roughness to "be investigated the constant B of the velocity distrlhution law, namely: B = — - 5.75 log I (I6.if5) ^#r ^ By comparison of equation (iS.k^) with (16.40) one ohtalns for the equivalent sand roughness-: ]£ 5.75 Zog -^ = 8.5 -B ' (16. 46) k In this way one may determine the drag for arhitrary roughnesses from a simple measurement in the roughness tunnel. This jaethod may "be also carried over to the case of the intermediate region. CHAPTER XVTI. THE FRICTION DRAG OF THE FLAT PLATE IN LONGrmOINAL FLOW The tur"bulent friction drag of the plate in longitudinal flow is of very great practical importance, for Instance as friction drag of wlngs^ airplane fuselages, or ships. The exact measurement of the friction drag for the large Reynolds numhers of practice is extremely difficult. Thus WACA TM No. 1218 31 it is particularly important that one can, according to Prandtl (references 73 bxiA. 7^)? calculate the friction drag of surfaces from the results of pipe flow studies. This conversion from the pipe to the plate can he made for the smooth as well as for the rough plate. a. The Smooth Plate One assumes, for simplification, that the boundary layer on the plate is turbulent from the leading edge. Let the coordinate system he selected according to figure 86. The boundary layer thickness 6(x) increases with the length of run x. Let b be the width of the plate. For the transition from pipe to plate the free stream velocity Uq of the plate corresponds to the maximum velocity u,^ in the pipe, and the boundary layer thickness 6 to the pipe radius r. One now makes the fundamental assumption that the same velocity distribution exists in the boundary layer on the plate as in the pipe. This is pertalnly not exactly correct since the velocity distribution in the pipe is influenced by a pressure drop, whereas on the plate the pressure gradient equals zero. However, slight differences In velocity distribution are insignificant since it is the momentum Integral which is of fundamental Importance for the drag. For the drag W(x) of one side of the plate of length x, according to equations (lO.l) and (10.2) ¥(x) = b T (x) dx = bp p5(x) u (Uq - u) dy (17.1) whence 1 dW b dx ^o(x) (17.1a) The equation (I7.I) can also be written in the form il W(x) = bpu/6(x)j -^ ^1 - -^j d i (17.2) For the velocity distribution in the boundary layer one now ass^jmes the 1/7-power law found for the pipe. Replacing u^^^ by U^ and r by 5]^ one may write this law, according to equation (16.II): 32 NACA TM No. 12l8 = V 1/7 (17.3) Hence the momentum. Integral "becomes il 01 J = u 1 - t) d i s .''% l/7\ Tl« dV ^ (17A) and thus W(x) = ^ hpTj/5(x) Hence follows^ according to equation (17.1a) (17.5) ° - 72 P^o dl (17.6) On the other hand, one had found "before for the smooth pipe, equation (16.I3) again replacing r "by 5 and u "by U : TTWY " o Tq = 0.0225 pUo'^^ (^^' (17.7) By equating equations (17.6) and (17.7) results: 7 2d5 "^A/vN^A 12 P^o i = °-°225 PTJ, (5 This is a differential equation for 5(x). The integration yields; k 5A 72 /v\^A _5 =-0.022,1-) X > (17.8) / X / v\l/5 i^/5 6(x) = 0.37(^1 ' X ^^ MCA TM No. 1218 33 5(x) _ 0.37X Ee, V (17.9) For the turliulent "boundary layer the boundary layer thickness, therefore, increases with x^/5. The corresponding equation for the laminar flow was, according to equation (9.21a), 6 = 5\/vx/Uq. By suhstitution of equation (17.8) into equation (17. 5) one ohtains 2 -^/5 W(x) = 0.036 hpUg X (Re^) or for the drag coefficient c„ = W/^ U xb: Cj. = 0.072 (Ee ) -1/5 Comparing this result with test results on plates one finds the numerical value 0.072 to he somewhat too low. c„ = 0.074 (Ee,^) -1/5 valid for 5 X lo5 < Ee < 10''' (17.10) corresponds "better to the measurements. This law holds true only 7 for Ee < 10 , corresponding to the fact that the Blasius pipe resistance law and the l/7— power law of the velocity distri'bution, which form the hasis of this plate drag law, are not valid for large Eeynolds numbers. This law is represented in figure 87 together with the laminar— flow law according to equation (9.19)« The initial laminar flow on the front part of the plate can he taken into consideration by a subtraction, according to Prandtl (reference 73): c-f = 0.07ij- Ee;j -1/5 1700 Ee 5 X 105 < R < 10 7 (17.11) The plate drag law for very large Eeynolds numbers can be obtained in essentially the same way by starting from the universal logarithmic law for the velocity distribution equation (l6.l4-l) which, according to 3h NACA TM No. 12X8 its derlTatioiij is valid up to Eeynolds numbers of arbitrary magnitude. Here the calculation TDecoines considerably more complicated. The development of the calculation is clarified if one first introduces the Telocity distribution in a general form. We had introduced for the pipe y^* The values at flow the dimensionless variables cp = -^ and ti the edge of the boundary layer are to be denoted by the index 0, thus ^o = 5v. ^o = (17.12) Then: ^ = ^*^ = ^o ^ (17.13) y = dy V^o U dT, (17.14) From the equation t = follows, with W according to equation (17.1) o b dx P ^* P dE <> ' ^(u. - ^) dy ;> r 05 ^^, \^ =T^T-< ^(Uo - u) dy dx dTio ''o and according to equation (17.13) U_ vcp o d 9o 2 " ^o ^% u(Uo - u)dTi dx NACA TM Wo. 1218 35 o _ Jo _d_ (p^2 " ^° dx dTlo ,no t^-t'^^ ^ " dX dTl^ ^O V "Pr dn U 6.1] _Q_ = V —a -d— cp^2 dx dri^ cp-^)d, (17.15) 2\ In forming the integral -= — P f cp — ^ j drj one must note the following facte: The differentiation with respect to the upper limit gives zero, since for t\ = t)q, cp = cpQ. In the differentiation of the integrand 9 is to he regarded as constant, and cp as a function of t) . Therefore •iTlo rih ^1o Tie 9 dT) 9 Thus follows from equation (17.15) 1% dx dT), (17.16) One puts, for simplification. dcp^ 1% F(T,^) = ^ I 9% (17.17) and obtains from equation (17.16) 36 NACA TM No. 12l8 ^ = xf ^^^o) '^\ If one assijines this law to "be valid from ths leading edge of the plate (x = 0), that iSj that the flow is turtulent starting from the front, the integration gives: X = X , (,^) (17.18) with 1^1 (m) = ^^^o)^ (17.19) 'ao=° Equation (17. l8) can also he written so that the Ee-mxmber formed with the length of run x appears: U X Ee^ = -^ = $ (r,^) (17.20) with: (Til) dcp '%=° R^. 9 dTi dTi ,"o (17.19a) Equation (17.20) gives the relation hetween ths dimensionless houndary layer thickness t^ = v^b/v and the Re-number U x/v . The drag remains to "be calculated. From follows J "because ^ - \> P r ^ dji NACA TM No. 12l8 37 1^1 ¥ = -b p Uo V F(Tlo) Jt1q=o ^r dTiQ = b p Uq vt (ti^) (17.21) where t (tI^) P^l F(no) ^, dTi 2 'o n.=° (17.22) /p 2 The drag coefficient c = ¥/^ Uq "b x "becomes finally c. = l^i, . = 2 ^ V (Til) «(t1i) (17.23) Hence C|. also turns out to he a function of t]-,. Equations (17.20) and (17.23) together give a parametric representation of c as a function ^ v„5 of Ee-jj-j where the parameter is the boundary layer thickness r\ = — ^— . Numerical Results In order to arrive at numerical results, one must introduce a special function for 9(1)). For the l/7— power law according to equation (16.11) that is, with cp = 01]/', one would obtain the drag law according to equation (17.IO). One uses the universal logarithmic velocity distri- bution law, equation (l6.l4). cp = 2.5 In T] + 5.5 In order to make the carrying out of the integrations more convenient, one writes cp = 2.5 In (1 + 9ti) Then 9 becomes, for t) = 0, cp = 0. The adding of the one changes cp(ti) a little, only for very small r\ and has only litble influence on the integrals. If one writes the law at first in the general form 38 • NACA TM No. 12l8 cp = a 2n (1 + "bT)) (17.21+) the calculation of the Integrals equations (17.17)^ (17. 19)^ (17.22), with z = 1 + Tdt] yields F(t]) = a^ (zn^z _2toz + 2-- ^(t) = — (2 2n z — kz Zn z — 2 In z + 6z — 6 j ^^) = =h S^ + 1 - 2(z -1) 2n z > (17.25) With the numerical values a =2.493 t = 8.93 one ohtains for the drag lav the following tahle: 103 -6 Ee 10 c 10^ f 0.500 0.337 5.65 1.00 0.820 J+.75 2.00 1.96 i+.05 3.00 3.25 3.71 5.00 6.10 3.3i^ 12.0 17.7 2.81 20.0 32.5 2.57 50.0 96.5 2.20 100.0 217.5 1.96 500.0 li+Ol.O 1.55 This tatle can he replaced by the following interpolation formula: Cj = 0.472 (Zog Ee^.) 2.58 NACA TM No. 12l8 39 Comparison -with test resiilts shows that the agreement improTes if the number 0.472 is slightly varied, hy putting °f = O.k^^ (log Ee^) 2.58 6 9 10 < Ee^< 10^ (17.26) Prandtl - Schlichtlng's Plate Drag Law The laminar approach length may again "be taken into consideration "by the same sobtraction as before; thus: Cf = OA55 _ ITOO (log Ee^) ^ ^ 5 X lo5 < Ee^ < 10-- (17.27) Whereas the system of formulas equation (17.25) Is valid up to Ee-^iumbers of arbitrary magnitude, the interpolation formulas, eq^uations (I7.26) and (17.27), have the upper limit Ee = 10 . However, this limit taJces care of all Ee— numbers occurring in practice. The theoretical formula (equation (17.27)) is also plotted in figure 87. Figure 88 gives a comparison with test results on plates, wings, and airship bodies. The agreement is quite good. Very recently this plate drag law has been somewhat improved by Schultz-Grunow (reference 89). Until then, the turbulent velocity profile measured in the pipe (l/7— power law, logarithmic law) had been carried over directly to the plate, mainly because accurate velocity distribution measurements of the plate boundary layer did not exist. The exact measurement of the plate boundary layer showed, however, that the plate profile does not completely coincide with the pipe profile. The teat points show, for large distance from the wall, a slight upward deviation from the logarithmic law found for the pipe. Thus the loss of momentum on the plate is somewhat smaller than that calculated with the logarithmic law. Schultz-G-runow repeated the calculation of the drag law according to the formula system given above with the velocity distribution law for the plate measured by him. His result Is represented by the interpolation formula °f = O.lj-27 (- 0.1+07 + ^og Ee„) 2.61+ 6 10 < Ee, < 10' (17.28) I^Q NACA TM No. 12l8 This law Is also plotted in figure 87. The differences from the Prandtl- Schlichting law are only slight.* The corresponding rotationally— symmetrical problem, that is, the turbulent boundary layer on a body of revolution at zero incidence, was treated by C. B. Millikan (reference 79). The l/7-power law of the Telocity distribution was taken as basis. Application to the general case has not yet been made. Twelfth Lecture (February 23, 1914-2) b. The Rough Plate The conTersion from pipe resistance to the plate drag may be carried out for the rough plate in the same manner as described previously for the smooth plate. One assumes a plate uniformly covered with the same rough- ness k. Since the boundary layer thickness 5 increases from the front toward the rear, the ratio k/5 which is significant for the drag decreases from the, front toward the rear. Behind the initial laminar run, therefore, follows at first the region of the fully developed roughness flow; the so- called Intermediate region follows and farthest toward the rear there is, finally, if the plate is long enough, the region of the hydraullcally smooth flow. These regions are determined by specification of the n-umerical values for the roughness coefficient v^k /v. In order to obtain the drag of the rough plate, one must perform the conversion from pipe flow to plate flow for each of these three regions individually. This calculation was carried out by Prandtl and Schlichting (reference 76), based on the results of Nikuradse (reference 71) for the pipe tests with sand roughness. For this conversion one starts from the universal velocity distribution law of the rough pipe according to equation (16.28), the quantity B being dependent also on the characteristic roughness value v^k /v^ according to figure Qk. The calculation takes basically the same course as described in detail for the smooth plate in chapter XVTIa. It is, however, rather complicated and will not be reproduced here. One obtains as final result for the total drag coefficient of the sand— rough plate a diagram (fig. 89) • which represents the drag coefficient as a function of the Reynolds number U^Z/v with the relative roughness Z/kg as parameter. Just as for the pipe, a given relative roughness 2 /kg has a drag Increasing effect not for all Ee-numbers, but only above a certain Re-Hiumber. This diagram is applicable also for roughnesses other than sand roughness, if one uses the equivalent sand roughness. In the diagram (fig. 89) the square drag law is attained. Just as for the pipe, for every relative * The tables pertaining to the plate drag formulas are given in table 7 5 chapter XXII. NACA TM No. 12l8 kl roughness 2 As provided the Ee-number is sufficiently large, interpolation formula The Cj = [1.89 + 1.62 Zog ^ j 2 A-2.5 (17.29) applies to this law. c. The Admlssitle Eoughness The problem of the admissible roiighness of a wall in a flow is very important in practice since it concerns the effort that might reasonably be expended in smoothing a surface for the purpose of drag reduction. Admissible roughness signifies that roughness above which a drag increase would occur in the given turbulent friction layer (which, therefore, still is in effect hydraulically smooth). The admissible relative roughness kg/z decreases with increasing Re— number UqZ/v as one can see from figure 89. It is the point where the particular curve Z/kg diverges from the curve of the smooth wall. One finds the values for the admissible relative roughness according to the following table; they can also be combined into the one formula ^o ■'^s admlss. 102 (17.30) U I V lo5 xo« 10^ 108 105 admlss. 10-3 10-^ 10-5 10^ 10-^ From equation (17.3O) one recognizes that the admissible roughness height is by no means a function of the plate length. This fact is significant for instance for the admissible roughness of a wing. Equation (17.3O) states that for equal velocity the admiesible roughness height is the same for a full scale wing as for a model wing. Let us assume a numerical example : Wing: chord 2 = 2m Velocity U = 3OO km/h = 83 m/sec k2 KACA TM No. 12l8 From equation (17. 30) results an admissible magnitude of roughness kg = 0.02 mm. This degree of smoothness is not always attained "bj the wing surfaces manufactured in practice, so that the latter have a certain roughness drag. In the considerations just made one deals with an increase of the friction drag in an a priori turbulent friction layer. However, the roughness may also change the drag "by disturbing the laminar friction layer to such an extent that the point of laminar/ turbulent transition is shifted toward the front. Thereby the drag can be increased or reduced according to the shape of the body. The drag is increased by this displacement of the transition point" if the body in question has predominant friction drag (for instance wing profile). The drag might be reduced, circumstances permitting, for a body with pre- dominant pressure drag (for instance, the circular cylinder). One calls the roughness height which causes the transition the "critical roughness height". According to Japanese measurements (reference 77) this critical roughness height for the laminar friction layer is given by ^*^crlt * -^^^ = 15 (17.31) A numerical example follows: Assume, as prescribed before, a wing Z = 2m U = 300 km/h = 83 m/sec then Re = UqI/v = lo'^. Consider the point of the wing x = 0.12, thus Re^j. = U x/V = 10 . Up to this point the boundary layer might remain laminar under the effect of the pressure drop. The wall shearing stress for the laminar boundary layer is according to equation (9.17) Tq ^ ^^^ ^^ 2.rir" 6900 m^ -^ = 0.332 Uq \^-^ = 0.332 -^— = 2.29 p P VUo^ 103 sec^ hence: T* = V^o/P = 1*52 m/sec and according to equation (17.3I) k = 15 T^ = -^^- i 10'~'^ = 0,14 mm crit ^* 1.52 7 NACA TM No. 12l8 k3 The critical roughness height causing the transition Is^ therefore^ ahout ten times as high as the roughness height admissible In the turhulent friction layer. The laminar friction layer therefore "tolerates" a greater roughness than the turhulent one. The following can he said ahout the Influence of the roughness on ^the form drag: Sharp— edged "bodies are Indifferent to surface roughnesses "because for them the transition point Is fixed hy the edges, as for Instance for the plate normal to the flow. Short ciirved bodies, on the other hand, as for instance the circular cylinder, are sensitive. For the circular cylinder the critical Reynolds number, for which the known large pressure drag reduction occurs, is largely dependent on the roughness. With increasing relative roughness k/R (E = radius of the circular cylinder) Ee^j^j^^ decreases. According to British measurements (reference 9O) "tiie drag curves for a circular cylinder with different relative roughnesses have a course as indicated in figure 90. The boundary layer is so disturbed by the rot^hness that the laminar /turbulent transition occurs for a considerably smaller Ee-number than for the smooth cylinder. The roughness has here the same effect as Prandtl's trip wire, that is, in a certain region of Ee-^iumbers it decreases the drag. It is true, however, that the supercritical drag coefficient is then always larger for the rough circular cylinder than for the smooth one. CHAPTER XVTII. THE TUEBULENT ERICTICJN LAYER IN ACCELEEATED AND RETARDED FLOW The cases of turbulent friction layer treated so far are relatively simple insofar as the velocity outside of the friction layer along the wall is constant. Here as for the laminar flow the case of special interest is where the velocity of the potential flow is variable along the wall (pressure drop and pressure rise). As for laminar flow, the form of the boundary layer profile along the wall is variable. In practice this case exists for Instance for the friction layer on the wing, on the turbine blade, and in the dlffuser. Of special interest is the question of whether separation of the boundary layer occurs and, if so, where the separation point is located. The problem consists therefore for a prescribed potential flow in following the turbulent friction layer by calculation. The calculation of the turbulent friction drag is of Importance. The corresponding problem for the laminar friction layer was solved by the Pohlhausen method (chapter X). For the turbulent friction layer the method of Gruschwitz (reference 78) proved best. Gruschwitz makes the asumption that the velocity profiles of the turbulent boundary layer for pressure drop and hk NACA TM No. 12l8 rise can be represented as a one— parameter family, if one plots u/u against y/-a. -d signifies the momentum thickness which is, according to equation (6.32), defined "by: U^^ =1 u(U - u) dy (18.1) As form parameter one selects (18.2) u('3) denoting the velocity in the friction layer at the distance from the wall J = -a. That t[ actually is a serviceable form parameter can he recognized from figure 9I where a family of turbulent boundary layer profiles is plotted according to Gruschwitz. Gruschwitz found from his measurements that the turbulent separation point Is given by Ti = 0.8 (Separation) (I8.3) The form parameter r] is analogous to the Pohlhausen— parameter X of the laminar friction layer. However, a considerable difference exists between -q and X: for the laminar friction layer an analytical relation exists between X and the press''jre gradient and the boundary layer thicknessj namely according to equation (10.41) g2 X = -^^ (18. i^) V dx Such a relation is thus far lacking for the t^lrbulent boundary layer, since one does not yet possess an analytical expression for the turbulent velocity profiles*. One needs therefore an empirical equivalent for equation (l8.J|). For the special case of the turbulent friction layer without 1/7 pressure gradient where the l/7— power law u/U = (y/5) applies for the velocity profile, one finds from equations (I8.I) and (l8.2) * Compare, however, chapter XXIIb, where under certain assijmptions such an analytical connection is indicated. NACA TM Wo. 1218 i^5 1 6 J_ 72 ^ = -L T) = 0.1+87 (18.5) Since in the case of the turhulent "boundary layer, an analytical expression for the velocity dlBtrnDutlon does not exist, the calculation is limited to the determination of the four characteristics of the friction layer: form parameter t], wall shearing stress t displacement thickness 5*, momentum thickness t3 . Four equations are required for their calculation. As for the laminar 'boundary layer, the momentum theorem yields the first equation; the momentum theorem may, according to equation (10. 36) be written in the form: (I) (18.6) The second equation is yielded "by the function T = ^^^) (II) (18.7) obtained by Gruschwitz by evaluation of the measured velocity profiles (fig. 92) J and regarded as generally valid. It can be derived also by calculation from the form of the velocity profile (compare appendix chapter XXIl) and yields: Tl = 1 - H - 1 H(H + 1) H-1 (18.8) The third equation is empirically derived by Gruschwitz from his measurements. He considers that the energy variation of a particle moving parallel to the wall at the distance y = i3 is a function of u^, U, t3, V. Dimension considerations suggest the following relation: ^^=F(„Re) q dx (18.9) = |n2 y = ^ and g-, = p + § u^^ signify the total pressure in the layer The evaluation of the test results showed that a dependence on 1^6 NACA TM Wo. 1218 the Ee— number is practically non-existent, and that one can represent equation (18.9) in the following manner: ■^ i = O.OO89U T] - O.OOlj-61 <1 djc (18.10) Furthermore, the identity go - V^^^-V P 2 2^^ = £u2fl-]£i i.ciT, if is valid. One puts 1 T] = ^ ■ (18.11) and has therefore dg-L dx dC dx (18.12) Now equation (I8.IO) can "be written: d^ ■a ^ = - 0.00894 ^ + 0.00i|6l q (III) (18.13) The fourth equation is still missing and is replaced by the following estimation of Tq: According to the calculations for the plate in longitudinal flow, equation (IT-T) was: ^ - 0.0225(— j = 0.0225 (Eeg)' -lA (18.14) If one takes into consideration that for the l/7— power law of the Telocity distribution: NACA TM No. 12l8 ^7 6.=|5, i3 = -1- 5 72 one can write equation (l8.lU) also; Tn -lA -lA ° - 0.01338 (Reg^) = 0.01256 (Ee^) 01/ (17) (18.15) For calculation of -a and t] (t, = Q. r\, respectlTsly) one must now solve the following system of equations: — + 0.00894 -^ = 0.01+61 i dx V V i^ ^ A ^ H\ .fl dg djc I 2y q djc PU^ (18.16) 2 T = •^17 is a given function of x; H and — ^ 2 pU^ are given functions of T] = ^/q or i3, respectively. This system of equations is to be solved downstream from the transition point. Initial values: As initial value for i3 one takes the value from the laminar friction layer at the transition point: i3 oturh. •a olam. (18.17) This is "based on the consideration that the loss of momentum does not vary at the transition point since it gives the drag. The initial value of r\ is somewhat arbitrary. G-ruschwitz takes T1_ = 0.1 and states that a different choice has little Influence on the result. With these initial values the system of equations (18.16) may he solved graphically, according to a method of Czuher (compare appendix. Chapter XXII, where an example is given) . A first approximation for i3 is ohtained by first solving the second equation with constant values for Tq /pU and H; 1^8 WACA TM No. 1218 L2. 2 PU = 0.002; H = 1.5 (18.18) are appropriate. Thsre'by the second equation Is a differential eq^uation of the first order for -d. This first approximation i3t (x) is then suhstituted into the first equation, vhich then 1)60011168 a differential equation of the first order for ^(x); let its solution te denoted hy L(x). Thus one has also a first approximation for tj: t]-,(x). With T|, (x) one determines the course of H(r|) according to figure 92 and is now atle to improve t according to equation (I8.I5). These values of hoth H and t are now inserted in the second equation, and a second approximation -Boi^) is obtained. By sutstitution of -SoCx) into the first equation one ohtains the second approximation ^p(x), etc. The method converges so well that the answer is essentially attained in the second approximation. The separation point is given ty Ti = .0.8 Incidental to the houndary layer calculation one obtains the following characteristic values of the friction layer as functions of the arc length X : ■bM, b*(x), Ti(x), To(x). The "boundary layer calculation for the profile J OI5, c^^ = is given as example in figure 93 • The transition point was assumed at the velocity maximum. The calculation of the laminar "boundary layer for the same case was Indicated in chapter XII. The details of this example are compiled in the appendix, chapter XXII. It should "be mentioned that the calculation for the tur'bulent "boundary layer must "be performed anew for every Ee-numier UqI/v, whereas only one calculation was necessary for the laminar "boundary layer. The reasons are, first, that the transition point travels with the Ee-number, and second, that the initial val ue of i3/t varies with Ee, since for the laminar boundary layer t- \ — ^=^ at the transition point Is fixed. "t y V It must be noted that the values obtained for Tq become incorrect in the neighborhood of the separation point: At the separation point t must equal zero, whereas equation (18.I5) gives everywhere t / 0. MCA TM No. 1218 kg B oundary Layer Without Pressure Gradient In this case q(x) = Constant. Equation (18.I3) can te written: 1 i^ = _ - ^ = - o.oo89ij-Ti + 0.001^61 (18.19) or, "because q.(x) = constant; ^ = iMl = q U (18.20) djc dx ^ djc Thus equation (18.I9) becomes; ^ ^ = - 0.0089UT1 + o.ooi)-6i (18.21) A solution of this equation is: . 0,00461 ^ g (^8.22) ' 0.0089!^ ^ ^ Since at the ■beginning of the tiirbulent friction layer t\ is smaller than this value (transition point t) = 0.1) and since according to equation (l8.2l) drj/djc > 0, t] must in this case approach the value T] = 0.516 asymptotically from "below. For the velocity profile of the 1/7— power law, t\ = O.U87 (compare equation (18.15)). The profile attained asymptotically for uniform pressure (p = constant) therefore almost agrees with the l/7— power law that was previously applied to the plate in longitudinal flow. A great many "boundary layer calculations according to this method are performed in the dissertation "by Pretsch (reference 80). Thirteenth Lecture (March 2, 19^12) CHAPTER XIX. FREE TURBULENCE a. General iRemarks; Estimations After considering so far almost exclusively the turbulent flow along solid walls, we shall now treat a few cases of the so— called free turbulence. By that one understands turbulent flows where no solid walls 50 WACA TM No. 12l8 are present. Examples are the spreading of a Jet and its mixing with the surrounding fluid at rest; or the wake flow behind a "body towed thi^ough the fluid at rest (fig. 9^)- Qualitatively these turbulent flows take a course similar to that for the laminar case (compare chapter IZ) ; quanti— tatively, howerer, considerable differences exist, since the turbialent friction is very much larger than the laminar friction. In a certain way, the cases of free turbulence are, with respect to calculations, simpler than turbulent flows along a wall, since the laminar sublayer is not present and the laminar friction as compared with the turbulent one can therefore be neglected for the entire flow domain. The free turbulence may be treated satisfactorily with Prandtl's concept of the turbulent shearing stress according to equation (I5.9): .1^ du Su , — (19.1) oy the mixing length I being assumed a pure position function. The turbulent friction has the effect of making the Jet width increase and the velocity at its center decrease with increasing distance along the Jet. We now perform rough calculations, according to Prandtl (reference 2, Part I), for a few cases of free turbulence which give Information about the laws governing the increase of width and the decrease of "depth" with the distance x. It has proved useful for such turbulent Jet problems to set the mixing length I proportional to the Jet width b: 1 = p = constant (19.2) b Furthermore, the following rule has held true: The increase of the width b of the mixing zone with time is proportional to the fluctuation of the transverse velocity v' : ^~v. (19.3) , D S S D/Dt signifies the substantial derivative: thus: — = u ^r— + v :r— . Dt dx dy According to our previous estimation, equation (I5.5): v' = 2 Therefore : MCA TM No. 1218 51 Dt dy Furthermore, the mean value of ^ equals approximately: - number — =^ Sy and thus: =rr = nuuiber t- u = number B u (I9.5) Dt D max max v ^ v; Jet (Plane and Circular] We shall estimate , "by means of these relations, how the width increases with the distance x and the velocity at the center decreaeea. At first, for the circular as well as for the plane Jet: =rr = number u -r- (19 -o) Dt max dx \ ^ j It follows, "by comparison with equation (I9.5): -T— = number r- = number dx b b = number x + constant If the origin of coordinates is suitably selected (it need not coincide with the orifice) one has therefore: b = number x (plane and circular Jet) (I9.7) The relation between u^^y and x Is obtained from the momentum theorem. Since the pressijire is constant, in the x— direction, the x— momentum must be independent of x, thus: J-p/u^dF = constant 52 NACA TM Nc 12l8 whence follows for the circular jet: or; 2 2 J = nuniber p u b u = niniber t- \| — max ^ \ P and "because of equation (I9.7) ^^max = number X Yp (circular Jet) (19.8) For the plane Jet, if J' signifies the momentum, per unit length of the Jet: J' '= numher p u h 1 y J' "^^ = ^™*er — \) -p- y^ and because of eq.uation (I9.7): (plane Jet) (19-9) Wake (plane and circular) The calculation for the wake is somewhat different, since the momentum which gives directly the drag of the "body must he calculated in a slightly different way. The momentum integral is now (compare equation (9.14.0)) : W=J=p/u(Uo-u)dF (19.10) At large distance from the body u* = Uo — u is small compared with U(- (fig. 95) so that u» « Uq and NACA TM No. 12l8 53 u(Uo - u) = (Uq - u' ) .u» Z Uq u« Thus one obtains for the circiilar wake: § c,., F U^2 «, p TJ u« jt h2 2 w ■" "'o t c F u* w — — /w — ^^— o no (19.11) Instead of equation (19.6) now applies: Dt ° dbc (19.12) and instead of equation (19.5) =rr = number t- u' Dt D (19.13) Equating of equations (19.12) and (19.I3) gives: tJ ^ ~ i u' = p u» o djc l3 (19.1^) By comparison with equation (I9.II) one obtains: dx jt p c^ F X (Wake Circular) (19.15) By insertion in equation (I9.II) results: u' .. 1 /"v ^ ^o 4p2,2 1/3 (Wake circular ) (19.16) '^h liACA TM No. 1218 For the plane wate "behind a long rod, wing, or the like with the diameter d and the length L,W = c — U^, Ld and ¥ = J^pUqU'13L and hence; c d (19.17) and from this in combination with equation (I9.IU) 2 h — ~ p c d dx w h ~ \fpTT7 (wake plane) (19.18) By substitution in equation (I9.I7) results; ^c d^ ^1/2 u' 1 w /V — — Uo 2 ^p ^/ / (Wake plane) (19.19) ThuSj for the circular wake, the width increases with \^y^, and. -2/3 the velocity decreases with x ; for the plane wake, the width increases with \f^, and the velocity decreases with x ' . The power laws for the width and the velocity at the center are compiled once more in the following tahle. The corresponding laminar cases which were partially treated in chapter IX, are included. More- over, the case of the free Jet boundary is given, that is the mixing of a homogeneous air flow with the adjoining air at rest. (Compare figure 97. ) NACA TM Wo. 1218 55 Lami nar TurlDulent Case Width Velocity at Center u max or u' respectively Width h Velocity at Center u max or u' respectively Plane Jet .2/3 X-V3 X x-1/2 Circular Jet X x-1 X x-1 Plane wake xl/2 x-1/2 xl/2 x-1/2 Circular wake xl/2 ^1 xl/3 x-2/3 Free Jet "boundary xl/2 x° X x° POWER LAWS FOE THE INCEEASE WUH WIDTH AND THE DECREASE OF TELOCITY WITH THE DISTANCE x For a few of the cases treated here the velocity distribution will he calctilated explicitly "below. The calculation on the "basis of the Prandtl mixing length theorem was performed for the free Jet "boundary, the plane Jet, and the circular Jet "by W. Tollmien (reference 81), for the plane wake "by H. Schlichting (reference 82) and for the circular wake "by L. M. Swain (reference 83). The equations of motion for the plane stationary case are, according to equation (lU.lU), if the laminar friction terms are completely neglected: = _i^ + i^+i ^Ih. ^ p ^ p Sx p Sy u ^ + V ^^ Sv Sv Sx hj Su Sv S^ 3y 1 M+ 1 ^+ 1 ^ p ^ p dx p Sy (19.20) y 56 MAC A TM No. 12l8 Td. The Plane Wake Flow \Je shall now calculate the velocity diatrihution for the plane wake flow. A cylindrical "body of diameter d and bt.zii h is considered. Further, let u' = U^ - u (19.21) he the wake velocity. One applies the momentum theorem to a control area according to figure 95 ^ "the rear ho-jndary B C of which lies at such a large distance from the hody that the static pressure there has the undisturhed value. As shown in detail in chapter H., equation (g.kO), one ohtains: W = h p / u (Uq - u) dy BC (19.22) ¥ = h p / (Uq - u' ) u' dy J BC For large distances "behind the "body u' « Uq so that one may 2 , . approximately neglect the term u» in comparison with 17^ u' m equation (19.22). Hence: (19.22a) Since, on the other hand W = c„ d h ^ U "", there "becomes: •' " 2 o i+h ^' iy = I c^ i u^ (19.23) -b This prohlem can "be treated "rfith the boundary layer differential equations; they read according to equation (I9.2O) with the Prandtl expression for the turhulent shearing stress according to equation (I5.9), with p and a^ neglected: SiACA TM Wo. 1218 57 ^ Su Su 1 St ^,2 Su S a u — + V — = = 22 TT Sx By ^ Sy 5u I 5v _ Q Sy By' Sx Sy / For ths mixing length one puts, according to equation (19.2) (I9.2J+) 2 = p b (19.25) Further : U' = U-, + Up + . ^ = h (19-26) (19.27) For the wake "velocity u-i and the width "b the power laws for the decrease and increase, respectively, with x were already foimd in equations (19.I8) and (19,19), One therefore writes: u -1/2 u ych^ ^^^) o \ w 1 (19.28) h = B (cwd x) 1/2 (19.20) According to equations (19.27) and (19.29) Sn _ 1 Bti ^y B(cwd x)^/2 ^^ It 2 X The estimation of the terms in equation (I9.2ij-) with respect to their order of magnitude in x gives: Bu -3/2 bv -3/2 Su -1 S u _ -3/2 ^:— ~x'; — ~x-^'; — ~x ; ~ x -^' ; Sx By ' Sy .2 V ~ X *The terms u^ • - • signify additional terms of higher approximation, which disappear- according to a higher power of x than does u-^. 58 WACA TM No. 1218 Hence the term v ^ ^ x whereas the largest terms ~ x . equation (l9.2i|-) is simplified to: 1 2 ^^1 ^ ^1 - U — - -^2 1 ° Sx ^ ^' Thus (19.30) The neglected terms are taken into account only in the next approximation. The further calculation gives: ^^1 / X \-l/2 S; c d _ i a f I - 1. f 2 X 2x SU' Sy = u. ^-1/2 \\i J f B(c^dx) 1/2 S^u ^y 1 n r-^V'^'f" ^ w ° ' c d ) ^2 B c d X w 2 Z 2 Bu-, S U-, 2 p^ U^^/^-^^^ ^ f'f" !/c c B/'c^dxj 1/2 After Insertion in equation (I9.3O) and eliminating the factor 2 1 / X \"^/^ / N U — (-rV) "the following differential equation results for f(Ti) i (f + Ti f) = ^ f«f" B (19.31) The houndary conditions are: Su y = h: u = 0; — - ■^ dy NACA TM No. 12l8 59 That is: Ti = 1: f = f • = (19.32) Th3 differential— eq^uatl on (I9.3O) may immediately te integrated once and gives: 2 ifTi= — f'^ + Constant 2 B Because of the hoondary conditions, the integration constant must equal zero; thus: 1 3 2 - f Ti = — f ' 2 ^ B . This may he Integrated in closed form: ^1/2 1/2 ^ \ |2pf df B dTi df J B ,^ , if V2P' 2fl/2 = \ pi 2 3/2 ^ c„^g^3^^ -^N5^"^^^' Because f = for ti = 1, C = - i J^ and hence: 3 y2p2 ,2 = l^(l_,^3/2) ■ (19.33) q _„2 \ / f = i 9 2p2 60 NACA TM Wo. 12l8 The condition f = for T] = 1 is simultaneously satisfied according to equation (19'33)« In. f", that Is — —, a singularity results at the Sy2 center (t) = O) and on the edge. For ti = 0, f" =oo; the velocity profile there has zero radius of curvature. At the edge there exists a discontinuity in curvature. In contrast to the laminar "boundary layer solutions, where the velocity asymptotically approaches the value of the potential flow, one obtains here velocity profiles which adjoin the potential flow at a finite distance from the center. The constant B remains to he determined: n+h Oh ph ni u' dy =*2 u' dy = 2 / u^ dy = 2U^ c^d B / f(Ti) dii J-h Jo t/o Jo From equation (19-33) one finds: 2 / (^ — 'H ) '^'H = iT^ UO and hence; and, by comparison with equation (19-23) or: b2 1 = i; B = \/lO p (19.3^) 2 2 Thus the final result for the width of the wake and the velocity distribution from equations (19-28) and (19.29) is: '^Eeviewer's note: Integrating from —1 to +1, as was done in the original German version, results in an imaginary term, which was avoided in the translation by integrating from to +1 and doubling the result. WACA TM No. 1218 61 "b = \ 10 (3 /"c d zS^'^ ^1 .. VlO / X \-l/2 2 (19.35) Uo 18P \c^^ 1-(J''^' The constant 3 = z/b is the only empirical constant of this theory; it must "be determined from the measurements. Comparison with the tests of Schlichting (reference 82) shows that the two power la^-s (equations (I9.28) and (I9.29)) are well satisfied, and also bhat the form of the velocity distrihution shows good agreement with equation (19.35)^ figia^e 96. The constant 3 is determined as p = 1 =,0.207 The solution found is a first approximation for large distancesj according to the meas'jrements it is valid for x/c^d > 50' For smaller distances -1 -1 /2 one may calculate additional -terms which are proportional to x , y: ' , . . . for the wake velocity in equation (19.26). The rotationally— symmetrical wake problem was treated by Miss L. M. Swain (reference 83). For the first approximation results exactly the same function for the velocity distribution; only the power laws for the width b and the velocity at the center u^' are different, namely b «' x ' and u^' ~ x ' , as already Indicated in equations (19.15) aJ^i (I9.16). Fourteenth Lecture (March 9, 19^2) c. The Free Jet Boundary The plane problem of the mixing of a homogeneous air streaiu with the adjoining air at rest shall also be treated somewhat more accurately (fig. 97). It is approximately present for instance at the edge of the free Jet of a wind tuiinel. The problem was solved by Tollmlen (reference 8I). The velocity profiles at various distances x are affine. One Gets u = U^f(Ti) = UJ'(ti) (10.^,6) 62 NACA TM No. 12l8 where Ti = |; (t~ x) (19.37) and nr\) =/f(n) ^T Furthermore set 2 = 01 = 01) (19.38) The equation of motion reads: u^+T^ = i^ = 2l2^^ (19.39) Sx dy p Sy Sy ^ 2 One integrates the continuity equation hy the stream function: ilf=j'udy=U^xj' f(Ti) dTi = U^ X F(ti) (I9.i^0) Then: |li = __£^F"; ^ = ^Y"; ^ = ^F'" ax X ' ' By X ' ^^2 ^2 T = -^ = -U^ (F - TiF') (19.J^0a) Suhatitution into the equation of motion (19.39) gi"^es, after division hy Uo^/x; FF" + 2c^ F"F'" = (19.^1) NACA TM No. 12l8 63 The boundary conditions are: at the inner edge: Tl = \' u = U : F» = 1 1^- F" = T = 0: F = Tl at the outer edge: T) = TI2: u = 0: F« = ^^ = 6: oy F" = > (19-^2) J Since the "boundary points r\^ and T)p are still free, these five hoimdary conditions can Tae satisfied by the differential equation of the third order equation (I9.J4-I). By introduction of the new variable: Tl* = ^y^ (19A3) the differential equation (19.41) is transformed into (' = differentiation with respect to t)*) F F" + F" F'" = (19.i^i^) The solution F" = 0, which gives u = Constant, is eliminated. The general solution of the linear differential equation F + F'" = (l9.i+5) is F = e \-x\> with \ signifying the roots of the equation X,-^ + 1 = 0, thus: 6U NACA TM Wo. 12l8 Hence the general solution is: F = r^ ^ VJ„J , . 2 ^^. /\fy = C^ e ' +0^6 cosl ^ T]*y + C e sin\^^ t]*; (19.^^6) If J moreover, one measitreB the ti— coordinate from the inner houndary point, thus puts: j^* = j^* - Ti*^ the solution (equation (I9A6)) can also "be written: Hi ... 21 =Tpf 2 F = d-| e + dp e cos f ^) . Ija 2 =1. (f V From the to-andary conditions (equation (I9.U2)) res'jlt for the constants the values: Tl* = 0.981 ; Tl* = - 2.01+ ; Tl* = - 3.02 d-L = -0.0062; d2 = O.987J d^ = 0.577 For the width of the mixing region one oh tains: t = ^[\ - \) = ^^2c2 ^^* - n|^ h = 3.02X^/20 2 X The constant c must he determined from experiments. From measurements it is found that h = 0.255 X (19.^7) MCA TM No. 1218 65 Hence ^ 2c^ = 0.08ij-5j c = 0.0174 and ^ = 0.0682 (19-i^8) It is striking that here the ratio z/b is essentially smaller than for the wake. The distrihution of the velocity components u and y over the width of the mixing zone is represented in figure 98. From the second equation of motion one may calculate the pressure difference "between the air at rest p ' and the homogeneous air stream p-, . One finds: p -p = o.ooJ+8 S u (19.49) i o 20 \ ^ ^J Thus an excess pressure of one— half percent is present in the Jet. For the inflow velocity of the entrained air one finds according to equation ( I9 . 40a ) : ^- CO = - FCrig) Uq = + 0.379 V 2c^ Uq and with the measured value of c; v_ „ = 0.032 Uq (19.49a) d. The Plane Jet In a similar manner one may also calculate the plane turbulent jet flowing from a long narrow slot (compare fig. 94). The laws for the increase of the width and the decrease of the center velocity have already been given in equations (I9.7) and (I9.9): "b «v Xj Uj^ ~ rj=- The calculation of the velocity distribution was carried out by Tollmien (reference 8I); it leads to a non— linear differential equation of the second order the integration of which is rather troublesome. Measurements for this case were performed by Forthmann (reference 91)- In figure 99 66 MCA TM No. 1218 the measurements are compared with "t^he theoretical curTe. The agreement Is rather good. Only in the neighhorhood of the velocity maximum is there a slight systematic deviation. There the theoretical curve is more pointed than the meas'Jired curve; the theoretical curve^ namely, again has at the maximum a vanishing radius of curvature. According to the Prandtl formula, equation (15.9)^ the exchange hecomes zero at the velocity maximum, whereas actually a small exchange is still taking place. e. Connection Between Exchange of Momentum, Heat and Material In concluding the chapter on turlsulent flows I should like to make a few remarks ahout the connection between the turbulent exchange and the heat and material transfer in a turbulent flow. In the Prandtl theorem equation (I5.9) for the apparent turbulent stress: T = p Z one can interpret: A P I Su Bu _ . Su kg sec I m (19.50) as a mixing factor. It has the same dimension as the laminar viscosity [x. Furthermore, the shearing stress t may be interpreted as a momentum flow: momentum T = = momentum flow 2 m sec (19.51) Momentum = mass X velocity = [kg sec] . Another effect of the turbulent mixing phenomena, besides the increased apparent viscosity by transport of momentijm,is the transport of all properties inherent in flowing matter, as heat, concentration of impurities, etc. If this concentration is not ■'jnlform, more heat or imparity is carried away by the turbulent exchange from the places of higher concentration than is brought back from the places of lower concen- tration. Thus there results, on the average, transfer from the places of higher to those of lower concentration. NACA TM No. 12l8 6? This results J for temperat-ure differences^ in a turliulent heat transfer; for concentration differences (for instance, of salt), in a turhiolent diffusion. They can, in analogy to eq^uation (I9.5O) be expressed as follows: ,, , „-. momentum transport „ d / moment omN Momentum flow = = A,- — I — —, 1 2 ' dv Vunit massy m sec '' Heat flow = heat transport = a^ ^ ( heat_\ 2 Q dy \unit mass/ m sec material N „, ri . . -, transport of material . d / Flow of material = = A, — ( ■ 2 ^ dy V cLy \unlt mass/ m sec The heat content of the unit mass is -c-e ( = temperatiire. c = specific heat cal m For chemical or mechanical 2 kg sec degree _ concentrations the concentration of material per unit mass is called the concentration c; it is therefore the ratio of two masses and therefore dimensionless. Thus the ahove equations may also be written in the following forms: T = A^ — T^dy > (19.52) , ^- \ dy The question arises as to whether A^, Aq, Aw are numerically the same or different. If the momentum is transported exactly like heat or material concentration — Prandtl's theorem is "based on this assumption — it would follow that A^ = Aq = Aj, and, for instance, the velocity and temperature distributions in a turbulent mixing region would have to be equal. However, measurements show partially different behavior. One has to distinguish between wall turbulence and free turbulence. Concerning free turbulence, calculations of G-. I. Taylor (reference 92) and measurements of Fage and Falkner (reference 93) showed for the velocity and temperature profile of the plane wake flow Aq -^ = 2 (free turbulence) (I9.53) At- 68 NACA TM No. 12l8 The heat exchange Is, therefore, larger than the momentum exchange. Conseq^uently the temperature profile is wider than the Telocity profile. The theory given for that phenomenon by G. I. Taylor operates with the conception that the particles, in their turbulent exchange moTements, do not maintain their momentum (Prandtl), hut their vortex strength 4^. (Prandtl's momentum exchange theory »« Taylor's vorticity transfer theory). However, there are cases not satisfied "by the Taylor theory (for instance the case of the rotationally symmetrical wake). That the heat exchange for free turbulence is considerably larger than the momentum exchange is also shown by experiments of Gran Olsson (reference 88) concerning the smoothing out of the temperature and velocity distributions behind grids of heated rod's. With increasing distance behind the grid the temperature differences even out much more rapidly than the differ- ences in velocity. For wall turbulence the difference between the mixing factors for momentum and temperature is smaller. H. Eeichardt (reference 8?) was able to show, from measurements of the temperature distribution in the boundary layer on plates in longitudinal flow by Elias (reference 86) and in pipes by H. Lorenz (reference Ik), that here A -^ = l.k to 1.5 (wall turbulence) (19. 5U) T Herewith we shall conclude the considerations of free turbulence. CHAPTER XX: DETERMINATION OF THE PROFILE DRAG FROM THE LOSS OF MOMENTUM The method, previously discussed in chapter IX, of determining the profile drag from the velocity distribution in the wake is rather important for wind tunnel measurements as well as for flight tests; we shall therefore treat it in somewhat more detail. The determination of the drag by force measurements is too inaccurate for many cases, in the wind tunnel for instance due to the large additional drag of the wire suspension; in some cases (flight test) it is altogether impossible. In these cases the determination of the drag from the wake offers the only serviceable possibility. The formula derived before in chapter IX, equation (9.^1) for determination of the drag from the velocity distribution in the wake is valid on.ly for relatively large distances behind the body. It had been assumed that in the rear control plane (test plane) the static pressure equals the pressure of the undistijrbed flow. However, in practically carrying out such tests In the wind tunnel or in flight tests one is NACA TM No. .1218 69 forced to approach the "body more closely. Then the static pressiire gives rise to an additional term in the formula for the drag. For measurements close behind the "body (for instance, for the wing, for x < t) this term is of considerable importance, so that it .must he known rather accurately. A formula was indicated, first hy Betz (reference 84), later hy B. M. Jones (reference 85) which takes this correction into consideration. Although at present most measurements are evaluated according to the simpler Jones formula, we shall also discuss Betz' formula since its derivation in particular is very interesting. a. The Method of Betz One imagines a control surface surrounding the Tsody as shown in figure 100. In the entrance plane I ahead of the "body there is flow with free— stream total pressure g^, hehind the "body in plane II, the total pressijre gp < g . The lateral boundaries are to lie at so large a distance from the "body that the flow there is undisturhed. In order to satisfy the continuity condition for the control surface the velocity U2 in plane II must he partially greater than the undisturhed velocity Ur Consider the plane prohlemj let the tody have the height h. Application of the momentum theorem to the control surface gives: W = h n+ 00 «+ 00 (^1 ^ ^\> - (^ u — 00 J — 00 Pg + Pu 2 ' *^ (20.1) In order to make this formula useful for test evaluation the integrals must he transformed in such a manner that the Integrals need to he extended only over the "wake". For the total pressures A at infinity: in plane I: In plane II: O O d O P 2 80 = Pi + 2 ^1 P 2 (20.2) / Outside of the wake the total pressure everywhere equals g^. Hence equation (20.1) becomes 70 NACA TM Wo. 1218 W = li 1+ CO (so - S2) dy + ^ u, — u^ ) dy (20.3) Thus the first Integral already has the desired fomij since the integrand differs from zero only within the wake. In order to give the same form to the second integral, one introduces a hypothetical substitute flow u.2'(y) ^^ plane II which agrees with Ug everywhere outside of the wake, Ug within the wake "by the fact that the total pressure g . Thus hut differs from for u„' equals + ^u». (20A) Since the actual flow Up satisfies the continuity equation the ^2_' "2 flow volume across section II for the hypothetical flow u.^ , u ' is too large. It shows a source essentially at the location of the "body which has the strength Q = ^J (^* - ^2) ^ (20.5) A source in a frictionless parallel flow experiences a forward thrust E = - P U„ Q (20.6) One now again applies the momentum theorem according to equation (20. 3) for the hypothetical flow with the velocity u at the cross section I and the velocity u'p at the cross section II. Since go = gQ and the resultant force, according to equation (20.6), equals R, one ohtains - P TT, oft = 2\ A Tin U •2> • (20.7) By subtraction of equation (20.7) from equation (20. 3) there results W + p Uq Q = h /(So - S2)^y ^ 2/(^'2' - -2') ^y (20.8) NACA TM lo. 1218 71 or because of equations (20. 5) and (20.6): W = h /(go - S2)^y ^ |/("'2' - ^2')^ - P ^o /(-'2 - -2X One may now perform each of these integrations only across the wake, since outside of the wake u» = u . Due to u'g^ - ug^ = C u» - u Vu» + u ) a transformation gives the following formula: (20.9) Betz' Formula In order to determine W according to this equation, one has to measure in the test cross section "behind the "body the following values: 1. Total pressure gj (therewith g is the value of gp outside of the wake). 2. Static pressure Pp. Furthermore, p = static pressure at infinity. Hence one obtains all quantities required for the evaluation of equation (20.9 ). It is useful for the evaluation of wind tunnel tests to introduce dimenslonless quantities. With F = ht as area of reference for the drag: V = c^ h t I u/ and hence from equation (20.9) (20.10) For the case in which p = p =0, at the test cross section one can 2 o write this equation, "because g^ = q^: 72 NACA TM No. 12l8 1 - ^^--JH-m (20.11) This agrees with equation (^.kl).* Thus in this case Betz' formula changes, as was to he expected, into the previous simple formula. h. The Method of Jones Later B. M. Jones (reference 85) indicated a similar method which in its derivation and final formula is somewhat simpler than Betz* method. Let cross section II (fig. 101) (the test cross section) lie close hehind the hody; there the static pressure p^ is still noticeably different from the static pressure p^. Let cross section I he located so far hehind the hody that the static pressure there equals the undis— turhed static pressure. Then there applies for cross section I according to equation (9.UI)* W = h p / ^1 (Uq - ^1) ^1 (20.12) In order to relate the value of u-, hack to measurements at cross section II, continuity for a stream filament is first applied: P ^1 iyn = P u„ dy^ (20.13) * In chapter IX the total drag of the hody (both sides of the plates) was designated by 2 Wj here the entire drag equals W! NACA TM No. 12l8 73 Jones makes the further assumption that the flow from cross section II to cross section I is without loss, that is, that the total pressure is constant along each stream line from II to I: So = Sn (20. Ill) First, according to equations (20.12) and (20.13); W = hp /u2(u^-ujdy. (20.15) Furthermore : P 2 p+— U =g=q ■^o 2 o °o ,^o Po + - ^■ P 2 2\ =h = ^2 P p p + — u '^ = go 2 2 2 with p =0 o > (20.16) and hence ^2 \ I ^2 ~ P2 U Un U_ (20.17) n P From equation (20.12) follows, with W = c t h ^ U^ : w -Sf^ ^l^ . ^2 ^r~ and hecause of equation (20.1?) (20.18) 74 NACA TM Wo. 1218 Formula of Jones Thus all quantities may te measured in cross section II close to the 'bod.j. This formula is simpler for the evaluation than Betz' formula, equation (20.10). In the limit, when the static pressure in the test cross section "becomes P2 = P , this formula, of course, must also transform into the simple formula equation (20.11). One obtains for po = p^ = from equat i on ( 20 . I8 ) : This is in agreement with equation (9.41). *• Fifteenth Lecture (March I6, 19lf2) CHAPTER ZXI: ORIGIN OF TURBULENCE a. General Remarks In this section a short summary of the theory of the origin of turhulence will "be given. The experimental facts concerning laminar/ tur"bulent transition for the pipe flow and for the "boundary layer on the flat plate have "been discussed in chapter XIII. The position of the tran- sition point is extremely important for the drag pro"blem, for instance for the friction drag of a wing, since the friction drag depends to a great extent on the position of the transition point. The so— called critical Reynolds nurriber determines transition. For the pipe (ud/v) . , = 23OO, and for the "boundary layer on the plate (Uqx/v) = 3 to 5 X 105. However, experimental investigations show the value of the critical Reynolds numher is very dependent on the initial distur'bance. The value of Re . , is the higher the smaller the crit ° initial disturbance. For the pipe flow the magnitude of the initial disturbance is given by the shape of the inlet, for the plate flow by the degree of turbulence of the oncoming flow. For the pipe, for instance, a critical Reynolds number (^<i/'^)crit ~ ^0,000 can be attained with very special precautionary measures. According to today's conception regarding the origin of turbulence, transition is a stability phenomenon. The laminar flow in itself is a solution of the Navier-Stokes differential equations up to arbitrarily high Reynolds numbers. However, for large Ee-^iumbers the laailnar flow NACA TM Wo. 1218 75 ■becomes unetatle, in the sense that small chance disturhances (fluctu- ations ,ln velocity) present in the flow increase with time and then alter the entire character of the flow. This conception stems from Eeynolds (reference 101). Accordingly, it ought to he posslhle to ohtain the critical Eeynolds number from a stahility investigation of the laminar flow. Theoretical efforts to substantiate these assumptions of Eeynolds mathematically reach rather far hack. Besides Eeynolds, Eayleigh (reference 102) in particular worked on the problem. These theoretical attempts did not meet with success for a long time, that is, no instability could be established in the investigated laminar flows. Only very recently has success been attained, for certain cases, in the theoretical calculation of a critical Eeynolds number. One assumtes for the theoretical investigations that upon the basic flow which satisfies the ITavier— Stokes differential equations a disturbance motion is superimposed. One then investigates whether the disturbance movement vanishes again under the influence of friction or whether it Increases with time and thus leads to ever growing deviations from the basic flow. The following relations will be intorduced for the plane case: basic flows: U(x, y)^. Y(x, y); P(x, y) disturbance movement: u'(x, y). v' (x, j) ; p'(x, y) resultant movement: U + u* . Y + v' ; P + p' > (21.1) J P, p* signify pressure. The investigation of the stability of such a disturbed movement was carried out essentially according to two different methods : 1. Calculation of the energy of the disturbance movement. 2. Calculation of the development of the disturbance movement with time according to the method of small oscillations. I am going to say only very little about the first method since it was rather unsuccessful. The second method was considerably more successful and will therefore be treated in more detail later. The first method was elaborated mainly by H. A. Lorenz (reference IO3). Thp following Integral expression may be derived for the energy balance- of the disturbance movement: ^ / EdV = p / MdY - 11 / NdY (21.2) 16 WACA TM Wo. 1218 In it E = -^ (u' + t' ] signifies the kinetic energy of the distur'baiLce movement. The Integration is performed over a space which participates in the movement of the basic flow and at the "boundaries of which the velocity equals zero, jrr signifies the substantial derivative. Thus one finds on the left side of equation (21.2) the increase with time of the energy of the disturbance movement. On the right slde^ M = - u' ,2 SU ax ,2 av v' + Sy u'v« + VSy > (21.3) N = vSx 3y / y The first integral signifies the energy transfer from the main to the secondary movement j the second the dissipation of the energy of the secondary movement. If the right side is greater than zero, the intensity of the secondary movement increases with time, and the basic flow is thus . unstable. An assumed disturbance movement u', v* satisfies merely the continuity equation, but no heed is paid to its compatibility with the equations of motion. If one could prove that the right side is negative for any arbitrao^y disturbance movement u*, v* this would serve as proof of the stability of the basic flow. On the other hand, the instability would be proved as soon as the right side is positive for a possible disturbance. Unfortunately general investigations in this direction are very difficult and have not led to much success. H. A. Lorenz (reference IO3) treated as an example the Couette— flow (fig. 102), assuming an elliptical vortex as a superimposed disturbance movement. He found for /U d\ this case V — tt— / = 288, whereas Couette' s measurements for this case ^ V /fcrit ' gave the value I9OO. b. The Method of Small Oscillations For the second method (method of small oscillations) the disturbance movement is actually calculated, that is, its dependance on the spatial coordinates x, y and the time t is developed on the basis of the hydrodynamic equations of motion. We shall explain this method of small oscillations in the case of a plane flow. In view of the applications of this method we shall immediately assume a special basic flow: the component U, namely, is to be dependent only on y and t and Y = 0. Such basic flows had been previously called "layer flows". They exist for instance in tunnel flow and pipe flow, approximately, however, also in the boundary layer since here the dependence of the velocity component XJ on the longitudinal coordinate x is very much smaller than the dependence on the transverse coordinate y. One now assumes a basic flow NACA TM Wo. 1218 U(y, t); Y= Oj P(x, y) 77 (21A) This basic, flow, ty itself, then satisfies the Navier-Stokes equations, thus ^ Su 1 5p S u — + = V — Bt P Sx ^^2 Sy = (21.5) / A disturbance movement which is also two— dimensional is superimposed upon this "basic flow:. disturbance motion: u»(x,y,t); T»(x,y,t); p*(x,y,t) (21.6) One then has as the resultant motion: u=U+u'; t=0+t*; p=P+p' (21.7) This resultant motion is required to satisfy the Navier— Stokes differential equations and one investigates whether the disturbance motion dies away or Increases with time. The selection of the initial values of the dis- turbance motion is rather arbitrary, but it must of course satisfy the continuity equation. The superimposed disturbances are assumed as "small", in the sense that all quadratic terms of the disturbance components are neglected relative to the linear terms. According to whether the dis- turbance motion fades away or increases with time, the basic flow is called stable or unstable. By insertion in the Navier— Stokes differential equations (3.I8) one obtains, neglecting the quadratic terms in the disturbance velocities 5U 5u» „ Su» — + + U + V St at Sx . Su 1 Sp 1 Sp' Sy P ax P Sx + A u' — + u — -— i ^* at ax p ay p ay ^ « V A V' >(2l.8) au» av« ax ay / 78 NACA TM No. 12l8 If one now notes the fact that the 'baBic flow by itself satisfies the Navier— Stokes differential eguationsj equation (21.5) j equation (21.8) is simplified to: Su' TT ^' , SU 1 Sp' ^ , + U + t' — + i— = V A u* Bt Sdc Sy P Sx St ^ P ay SU« , SV« _ Q -^ &vL+U^t1^1Se1^VAt» > (21.9) The pertinent "boundary conditions are: Tanishing of the disturbance components u* and t' on the bounding walls. From the system equation (21. 9) of three equations with three unknown quantities u*, t', p» one may at first eliminate p' by differentiating the first equation with respect to y and the second with respect to x and then subtracting the second from- the first. This gives, with continuity taken into consideration: S^u' ^ jj B^u' + ^, ^ _ 5^' _ jj SV BtBy + V SxSy a^u ,^ ^y ay^ StSx » S^u' S^t' Sy SxSy ax-^ Sx- > > (21.10) / In addition to this there is the continuity equation (21.9). There are now two equations with two unknown quantities u*, v*. Form of the Disturbance Movement For cases where the basic flow predominantly flows in one direction as for instance boundary— layer or pipe flow, the disturbance motion is assumed to be a wave progressing in the x-direction (= main flow direction), the amplitude of which depends solely on y. The continuity equation of the disturbance motion may in general be integrated by a disturbance function for which the following expression may be used: NACA TM No. 12l8 19 * (x^y^t) = cp(y)e ia(x-ct) (21.11)* Where: X = 2it/a the waTe length of the disturbance (a = real) C = C + 1 c. r 1 c = Telocity of wave propagation c^ = amplification factor; c . < 0: stable; c. > 0: unstable 9(y) = 9 (y) + i 9.(y) = amplitude of the disturbance movement r 1 From equation (21.11) one obtains for the components of the distirrbance movement li' = T^ = 9'(y)e oy St , , , ia(x^t) T» =-—- = - ioo:p(y)e dx > (21.13) / By substitution into equation (21.10) one obtains the following differ- ential equation for the disturbance amplitude cp: / i iafUcp" — ccp" - cpU" + a ccp - Ua cp j = v^cp"" - 2a cp" + a 9 j or (U - c)(9" - a29) - u"q) = =^-i^ (cp"" - 2aV + o-V) (21. U) Ow ■One introduces dlmensionless quantities into this equation by referring all velocities to the maximum velocity U of the basic flow (that is m for the friction layer the potential flow outside of the boundary layer and all lengths to a suitable reference length §) (for Instance, for ^he convenient complex formulation is used here. The real part of the flow function, which alone has physical significance, is therefore Re(t) C-it cp COS r a (- - V) cp. sin 1 a ( X — c t r , (21.12) 80 NACA TM Wo. 1218 the "boundary layer flow, the "bo-undary layer thickness). Furthermore, differentiation with respect to the dimensionless quantity y/5 will "be designated ty a prime mark ('). One then obtains from equation (21.1i|) (U - c)(cp" - a^qp) - U"cp = -^ (q)"" - 2aV + a\) (21.15) DISTURBADJCE Dlb'Jb'KHEETIAL EQUATION where E = ■■ ™ . This is the disturbance differential equation for the amplitude cp of the disturbance movement. The boundary conditions are, for instance, for a boiuidary layer flow y = (wall): u' = v' = 0: 9 = cp' = y = 00 : u' = "v* = 0: cp = qp' = (21.16) The stability investigation is an eigenvalue problem of this differential equation for the disturbance amplitude <p(y) in the following sense: A basic flow U(y) is prescribed which satisfies the Navier-Stokes differ- ential equations. Also prescribed is the Eeynolda number R of the basic flow and the reciprocal wave length a = 2it/x, of the disturbance movement. From the differential equation (21. I5) with the boundary conditions equation (21. I6) the eigenvalue c = Cp + i Cj_ is to be determined. The sign of the imaginary part of this characteristic value determines the stability of the basic flow. For Cj^ < the particular flow (U, R) is, for the particular disturbance a, stable; for Cj^ > 0_, unstable. The case Cj^ = gives the neutrally stable disturbances. One can represent the result of the stability calculation for an assumed basic flow U(y) in an a, R— plane in such a manner that a pair of values Cj., c± belongs to each point of the a, R— plane. In particular the : = in the a, R— plane separates the stable from the unstable curve disturbances. It is called the neutral stability curve (fig. IO3). In view of the test results one expects only stable disturbances to be present at small Reynolds numbers for all wave lengths a, unstable disturbances, however, for at least a few a at large Reynolds numbers. The tangent to the neutral stability curve parallel to the a-^axis gives the critical Reynolds number of the respective basic flow (fig. IO3). WACA TM No. 1218 81 Methods of Solution and Gteneral Properties of the Dlstiirban ce Differential Equation Since the stability limit (Cj^ = 0) is expected to occur at large Reynolds numbers^ it suggests itself to suppress the friction terms in the general disturhance differential equation and to ohtain approximate solutions from the so— called frlctionless disturbance differential equation which reads (U - c) (cp" - a^q)) -U"q) (21.17) Only two of the four "boundary conditions^ equation (21.l6),of the complete disturbance differential equation can now be satisfied since the friction- less disturbance differential equation is of the second order. The remaining boundary conditions are: y = 0: v« = Oj 9 = Oj y = 00: t» = 0: cp = (21. 18) The cancellation of the friction terms in the disturbance differential equation is very serious, because the order of the differential equation is thereby lowered from ij- to 2 and thus Important properties of the general solution of the disturbance differential equation of the fourth order possibly are lost. (Compare the previous considerations in chapter IV concerning the transition from the Navler— Stokes differential equations to potential flow. ) An Important special solution of equation (21.17) is the one for a constant basic flow, U = constant^ which is needed for Instance for the stability investigation of a boundary layer flow as a Joining solution for an outer potential flow. One obtains from equation (21.17) for U = constant: cp = e iay However, due to the boundary conditions for cp at y = ", the only permissible solution Is cp = e^ y (21.19) We shall prove at first two general theorems of Eaylelgh on the neutral and unstable oscillations of the frlctionless disturbance differential equation. 82 WACA TM No. 12l8 Theorem I: The wave Telocity c for a velocity profile with U"(y) — must, for a neutral oscillation (c^ =0, c = c ), egual the hasic velocity at a point so that there exists within the flow a point U — c = 0. Proof: (indirect) One makes the assumption c > U (= maximum, velocity of the hasic flow). One then forms from equation (21.17) the following differential expressions: 2 TT" L(cp) = cp" - a cp qp = (21.20a) U — c and 2- U" — L(cp) = cp" - a cp ^ cp = (21.201)) U - c L(q)) signifies the expression o"btained from 1(9) , if one inserts everywhere the conjugate complex q^uantities. Because of the "boundary conditions y = 0: cp = cp = y=oo. q3 = q) = o One forms further the expression 9L(cp) + cp L(cp) and integrates it "between the limits y = and y = co. The integrals may "be taien up to y = 00, since for large y, 9 »« e~^y. Because of equation (21.20aj b) J2_ must then "be ■^1 = cp L(q)) + cp L(cp) dy = (21.21) ^y=o After insertion of equation (21.20a and "b) results, "because c = c, J = j ( cp" cp" + cp"(p - 2a^cp9 - 2 — - — cp9 j dy = 'y=o or WACA TM Wo. 1218 83 noo J = 1 qxp' + cp'9 - 2 -"o llo cp'cp»dy - 2 I [ a^ + — ^^ ] cp9 dy = U - c The first term vanishes due to the houndary conditions, hence there remains Jl =-2 Jy=o cp»cp' +( a^ + — 2 — jqxp U - c / dy = (21.22) U" - cp'cp' as well as qxp are positive throughout; if U" — and c > U , U"/U' — c — and hence the integrand in eq^uation (21.22) is positive throughout. Thus the integral cannot "become zero. The assumption made at the "beginning c > U therefore leads to a contradiction. For "basic flows with "D" — 0, as for instance "boundary— layer flows in a pressure drop, the wave propagation velocity therefore must "be smaller than U^ for neutral disturbances. Hence a point IT — c = exists within the flow. This point is a singular point of the frictionless disturbance differential equation (21.17) and plays as such a special role for the investigation of this differential equation. The wall distance y at which U — c = is called j = Y-y- = critical layer. This first Rayleigh theorem proved a'bove applies — as shall "be noted here without proof — in the same manner to flows with tT" > 0. Sixteenth Lecture (March 23, 19^2) Theorem II: A necessary condition for the presence of amplified oscillations (c^ > 0) is the presence of an inflection point within the "basic flow (U" = 0). Proof: (indirect) According to assumption, c. ^ 0; thus U - c / for all y. With L(cp) and L(cp) one forms, according to equation (21.20a) and (21.20h), a simdlar expression as "before. This latter, integrated from y = to y = °°, must again give 0, thus noo J2 = [ 9 L(q)) - cp L(cp) dy = ]■ y=0 (21.23) 8U NACA TM No. 12l8 By substitution according to equation (21.20a and b) results with c = c — 1 c. r 1 '2 = '7=0 cpcp" - cptp" - ^(—^ 21. VU - c U - c dy = or '2 = W - qxp'' - 2i c. JQ u" cpcp dy = U - c {2±.2k) The first term again vanishes because of the boundary conditions. Since cp^ is positive thrcaghout and lu — c| / 0, the integral can only vanish if U" changes its sign, that is, an inflection point of the velocity profile U" = must be present within the flow. It has, therefore, been proved: In order to make the presence of amplified oscillations possible, an ■ inflection point mast exist in the velocity profile of the basic flow, or, expressed briefly, such oscillations are possible only for inflection point profiles. Later on Toll mi en (reference 110) proved that the presence of an inflection point is not only a necessary but also a sufficient condition for the existence of amplified oscillations. Hence the following simple statement is valid: Inflection point profiles are unstable. It must be mentioned that all these considerations apply in the limiting case B— ^<» since the proofs were obtained from the frictionless disturbance differ- ential eq_uation. ¥e know from our previous considerations about the laminar boundary layer that inflection point profiles always exist in the region of pressure rise, whereas in the pressure drop region the boundary layer profiles are always without an inflection point, (fig. lOij-). Hence we recognize that the pressure rise or pressure drop is of decisive significance for the stability of a boundary layer flow. The converse of the theorem just set up is also valid, namely, that for E — >oo velocity profiles without inflection point are always stable. From this, however, "one must not conclude that profiles without inflection point are stable for all Reynolds numbers. A closer investigation for Reynolds numbers of finite magnitude shows that there profiles without an inflection point also become unstable. One is faced with the peculiar fact that the transition from Re = » to Ee-number of finite magnitude, that is, the addition of a small viscosity to a frictionless flow, has a NACA TM No. 12l8 85 destabilizing effect, whereas one intuitively expects the opposite. As later considerations will show in more detail, the typical difference between the neutral stability curves of a basic flow with and without inflection point appears as represented in figure IO5. For the velocity profile without an inflection point the lower and the upper branch of the neutral curve have , for E — > 00 , the same asymptote a = . For the velocity profile with inflection point the lower and upper branch of the neutral curve have, for E — >», different asymptotes so that for E = 00 a certain wave length region of unstable disturbances exists. Furthermore the critical Eeynolds number is smaller for velocity profiles with an inflection point than for those without an inflection point. Hence it is to be expected for very large Ee-iiumber, to a first, very rough approximation, that the transition point in the boundary layer of a body lies at the pressure minimum. Figure IO6 shows schematically the pressure distribution for a rather strongly cambered wing profile at a small lift coefficient. The transition point would be expected in this case Just behind the nose on the pressure side, slightly more toward the rear on the suction side. Solution of the Disturbance Differential Equation In order to perform the actual calculation for the boundary— value problem Just formulated, one needs at first a fundamental system Cp-,, cp. °^ '^^^ general disturbance differential equation (21.15). One Imagines the basic flow U(y) given in the form of a power series development : U " U(y) = U' y + -^ y2 + . . . . (21.24) ° 2! If one introduces this expression into equation (21.15) and. then wants to construct a solution from the complete differential equation which satisfies the boundary conditions (equation (21. 16)), one encounters extreme difficulties of calculation, due to the two conditions to be satisfied for y = 00, In order to obtain any solution at all, one has to make various simplifications. The simplificationB concern: 1. The basic flow: Instead of the general Taylor— series equation (21.2^4-) one takes only a few terms, thus for instance a linear or a quadratic velocity distribution. 2. The disturbance differential equation: For calculation of the particular solutions the disturbance differential equation Is considerably simplified. Eegarding 1, it should be noted that linear velocity distributions frequently have been investigated with reepect to stability, as for 86 NACA TM No. 12l8 instance the Couette flow according to figiore 102 or a polygonal approxi- mation for cuTTed Telocity profiles according to figure 107. This facili- tates the calculation due to the fact that then the singular point U — c = is avoided in the frictionless dist'ur'bance differential equation (21.17) for neutral disturhances. However, all InvestigationB with linear velocity diBtritutions (references 104, 105, 106) were unsuc- cessful with the frictionless as well as with the complete differential equation. No critical Reynolds number resulted. When one later took for a "basis paraholic profiles, these negative results "became intelligl"ble. One must, therefore, take at least a parahollc distrl'bution as a "basis for the "basic flow. Regarding 2, it should "be noted that one can provide approximate solutions for the solutions of the complete differential equation (21.15) from the frictionless differential equation (21.17) since the solutions are required only for large Ee-^iumber E. The frictionless differential equation however can yield no more than two particular solutionsj two more have to "be calculated, taking the largest friction terms in equation (21.15) into consideration. The course of the calculation for the particular solutions will "be "briefly Indicated. One limits oneself to neutral dlstjir'bances, assumes a para'bolic velocity distrl'bution^ and imagines the latter developed in the neighborhood of the critical layer. y = yg.: U-c=U-c^ = U - c = U'^(y - y^) . !^ (y - j^f (21.25) The first pair of solutions 9-, j "Pp ^^ then o"btained from the friction- less dlstur"bance differential equation (21.17) "by suhstltution of equation (21.25). According to known theorems a"bout linear differential equations with a singular point a linearly independent pair of solutions has the form ^i=(^-^k)^(^-^e) u" (21.26) P, and P are power series with a constant term different from zero. The parttc-ular solution cp is especially interesting. NACA TM No. 12l8 87 9', -> °° for J = It, That Isj the u'— component of the dleturhance velocity becomes infinitely large in the critical layer. This can also he understood directly from the frictionlesB disturhance differential eq^uation (21.17). According to equation (21.17) cp" — a cp = U" U - c 9 or >p" j-^; ^' ~ ic8 (y - y^). If u"^ ^ This singular hehavior of the solution cp2 ^^ "the critical layer stems of course from neglecting the friction. The frictionless differential eq^uation here no longer gives a serviceable approximation. In the neighborhood of the critical layer the friction must be taken into consideration. Moreover, there is another inconvenience connected with the cpp' For fulfillment of the boundary conditions one req^uires the solution for y — y-g- > as well as for y — y„ < 0. However, for qpo it is at first undetermined what branch of the logarithm, should be chosen at transition from y — yg; ^ to y — y;g; "^ 0. This also can be clarified only if in the neighborhood of y — y^-, at least, the large friction terms of the complete differential equation (21.15) are taken into consideration. The details of the calculation will not be discussed here. The calcu- lation leads, as Tollmien (reference 109) tias shown, to the result that one obtains for the solution cp2 a so— called transition— substitution in the critical layer which appears as follows: U" 7-J^>0: cpg = P2(y -J^) + —(7- ^k) ^l(y - ^k) ^°e (7 - Y^ u" y - Jk < 0= ^2 = P2(y - ^k) ^ ^ (y - ^k) ^i(y - ^k) i°8 " y - y- K - in (21.27) ' If one writes, according to this, the complete u'— component, then in the neighborhood of 7 ~ J-^'- KACA TM No. 12l8 J - J^> 0: u' = < 0: u« U". K + — 1'°S (j - J-^) COB (ox - pt) .5^ .OS u" K cos (ax - pt) S (21.28) + It sin (ax — pt) y One ottalns therefore in the critical layer a phase discontinuity for the u* -component. This is retained eren in going to the limit, E — > 00 . it is lost, however, if one neglects the curvature of the basic flow U" or if one operates only with the frictionless differential equation. This phase discontinuity is very significant for the development of the motion. The loss of the phase discontinuity is the reason that stahility investi- gations neglecting the curvature U" or operating only with the frictionless differential equation remain unsuccessful. With this friction correction in the critical layer the pair of solutions cp 9. is sufficiently determined. By taking the friction '1' -^2 terms in equation (21.15) into consideration, one then obtains a second pair of solutions 9 , cp, which can he represented hy Hanlcel and Bessel functions. Of these -two solutions cp, tends very strongly towards infinity and is therefore not used "because of the "boundary conditions, equation (21. 16). cp tends, for large y, towards zero. The Boundary Yalue Pro"blem The general solution as a linear combination of the four particular solutions is: cp = C^cp^ + CgiPg + C^cp^ + C^cpj^ (21.29) Let us consider in particular the case where a "boundary layer profile is investigated with r-espect to sta"bility. For this case the "boundary value pro"blem can "be somewhat simplified. The previous considerations showed that in the disturbance differential equation the friction essentially needs to "be taken into consideration only in the neigh'borhood of the critical layer; also, of course, at the wall, because of non— slip. The critical layer is always rather close to the wall; hence for y > 6, NACA TM Wo. 1218 89 where U = U^j^ = constant^ one may use the frictionlees solution which is according to equation (21. I9) 9= e~°''^. Thus the condition that the solution for y = 5 Joins the solution for U = constant is 9' + a cp = $ = o 00 (21.30) This mixed "boundary condition is therefore to he set up on the outer edge. Furthermore, the particular solution cpi is a priori eliminated in the general solution (equation (21.29)), since it grows, for positive y — jy-, "beyond all limits j thus C. = 0. Hence there remains for the houndary •value according to equation (21.16) Ccp +Cq) +Cq) =0 1 lo 2 2o 3 30 C cp» + C cp' + C cp» =0 1 Id 2 2o 3 30 C$ +C<i) +C$ =0 ri5 2 26 3^35 ~\ > (21.31) y A further simplification takes place "because of the fact that "because of the rapid fading away of the solution cp on the outer edge y = 8, the solution $_ already practically equals zero. In the third equation of 15 equation (21. 3I) 35 may therefore he cancelled. The boundary value problem actually to he solved is, therefore. qp cp cp ^lo ^2o 30 cp* cp' cp* ^ lo ^ 2o ^ 2o 15 $^0 25 = (21.32) This determinant gives the eigenvalue problem indicated above, which requires - as has been said before - the solution of the following problem: Given 90 NACA TM No. 12l8 1. "basic flow U(y) 2. Eeynolds number Ee = U S/v 3. wave length of the disturhance a = 23t/a. One seeks from equation (21.32) the pertinent complex eigenvalue + 1 c . . Therein c gives the velocity of wave propagation and c. the anrplification or damping. 0=0 r Equation (21.32) may formally "be written in the foirTa: F(a, c^, c., E; TJ'^, U"^, ....)= (21.33) where equation (21.33) signifies a complex equation^ hence is equivalent to two real equations f fa, c , c., E; U' , U" , . . . = IV r 1 o o ' fJa, , c, E; U» , U" , . . . ) 2\~' r' 3.' ^ o-* o-* / > (21.3^) If one imagines for Instance Cp eliminated from these two equations, one obtains one equation "between a, E, c. : gJa,, c^, E; U»Q, U"^, . . J) = (21.35) From this equation Cj_ can be calculated as a function of a and E. The constants U*o, U"o, • • • are parameters of the basic flow. Thus, if equation (21.35) is assumed solved with respect to c.. i = gg^a, E; U'q, U"o, . . .j (21. 36) Cj = Finally one obtains from it, for the neutral disturbances c. = 0, a curve in the a, E— plane, given by the equation gg/a, E; U»^, ir^, . . .) (21.37) WACA TM No. 1218 91 This is the sought for neutral stahllity curve (compare figure IO3), which separates the unstable from the statle disturhances and also yields the theoretical stahility limit, that is, the critical Eeynolds number ^^crif The performance of the calculation, here only indicated, is analyt- ically not possible since the quantities a, c , E enter into the determinant, equation (21. 32), in a very complicated manner. One has therefore to resort to numerical and graphical methods. The critical Eeynolds number is Tery largely dependent on the form of the velocity profile of the basic flow, in particular on whether the velocity profile of the basic flow has an inflection point, thus on U"(y). The critical Eeynolds number found from such a calculation gives exactly the boundary between stability and instability, hence the first occurrence of a neutrally stable disturbance. In comparison with the transition point of test results it is therefore to be expected that the experimental transition point appears only for larger Eeynolds numbers where an amplification of the unstable disturbance has already occurred. c. Eesults A few results of such stability calculations will be given. The completely calculated example concerns the boundary layer on the flat plate in longitudinal flow with the laminar velocity profile according to Blasius (compare chapter IXa). In figure IO8 the streamline pattern of this plate boundary layer with the superimposed disturbance movement is given for a special neutral distvirbance. Figure IO9 shows, for the same neutral disturbance, the amplitude distribution and the energy balance. Since the distizrbance in question is neutral, the energy transfer from the main to the secondary movement is of exactly the same magnitude as the dissipation of the energy of the secondary movement. Figure 1^0 shows the neutral— stability curve as result of the stability calculation according to which the critical Eeynolds number is referred to the displacement thickness b* of the boundary layer (Uj^S'**'/v )„pj+ = 575- The connection between displacement thickness S* and length of run x is for the laminar boundary layer according to equation (9. 21) V* = 1.73 Thus a critical Eeynolds number formed with the length of run (U x/v) .^ = 1.1 X lo5 corresponds to the critical Ee-number (U 5*/v) = 575. The critical number for this case observed in tests ^ ' crit 92 NACA TM No. 12l8 was 3 to 5x10-^. It was explained atove tJaat it must be larger than the theoretical number. Furthermore^ figure 110 shows that at the stability limit the unstable wave lengths are of the order of magnitude X = 55- The unstable disturbances thus have rather long wave lengths. This calculation, carried out by Tollmlen (reference IO9) for the flow without pressure gradient was later applied by Schlichtlng (reference 11^4-, 115) "to boundary layer flows with pressure drop and pressure rise. The boundary layer profiles with pressure rise and pressure drop can be represented in a manner appropriate for the stability calculation as a one— parameter family with the form parameter X-p i^ according to Pohlhaueen's approximate calculation. One then obtains for each profile of this family a neutral— stability curve as Indicated in figure 111. Hence the critical Eeynolds number (^r^* / '^) crit ^^ ^ function of the form parameter X-p h according to figure 112. In retarded flow (X-p r <0) the critical Ee-number is smaller than for the plate flow (^pi^ = 0) , for accelerated flow {'>^-pu ^ 0) it is larger. With this result of a universal stability calculation the position of the theoretical transition point may be determined conveniently for an arbitrary body shape (plane problem) in the following manner: At first, one has to calculate for this body the potential flow along the contour, ftirthermore one has to carry out, with this potential flow, a boundary layer calculation according to the Pohlhausen method. This calculation yields the displacement thickness and the form parameter Xp^, as functions of the arc length along the contour, in the form - f^(s) and Xp^ = f2(B) Since in general there exists, accelerated flow at the front of the body and retarded flow at the rear X-nj. decreases from the front toward the rear. By means of the universal stability calculation according to figure 112 one may determine a critical Reynolds number i^-n^* / "^^ crit for each point of the contour. The position of the transition point for a prescribed Ee-^umber U^t/v is then given by the condition ^ ^crlf V V V /crit Figure 113 shows, for the example of an elliptic cylinder*, how to find the transition point. The curve (Uj^^-^/v)^^^^ decreases from the front *The boundary layer calculation for this elliptic cylinder was given in figure 52. , NACA TM No. 12l8 93 toward the rear; the curve Ujj^S^/v for a fixed Ee-^aumber U t/v increases from the front toward the rear. The intersection of the two curves gives the theoretical transition point for the respective Ee— number U-t/v. By determining this point of intersection. for various U-t/v one ohtains the transition point as a function of U t/v. The result is represented in figure 114. The transition point travels with increasing Ee-durober from the rear toward the front; however, the travel is considerahly smaller than for the plate in longitudinal flow which is represented in figure llij- for comparison. Finally figure II5 shows the result of such a stability calculation for four different elliptic cylinders in flow parallel to the major axis. The shifting of the transition point with the Ee— number increases with the slenderness of the cylinder. For the circular cylinder the shifting is very slight, which is caused hy the strongly marked velocity maximum. As a last result, figure II6 shows the travel with Ee— number of the transition point on a wing profile for various lift coefficients. The profile in question is a symmetrical Joukoweky profile with lift coefficients Cg^ = to 1. With increasing angle of attack the transition point travels, for fixed Ee-nijmber toward the front on the suction side, toward the rear on the pressure side. (compare the velocity distirbutions for this profile, given in figure 5^.) One recog- nizes that the shift of the transition point with the lift coefficient is essentially determined by the shift of the velocity maximum. The last examples have shown that it is possible to calculate beforehand the position of the transition point as a function of the Ee— number and the lift coefficient for the plane problem of an arbitrary body immersed in a flow (particularly a wing). Eegarding the comparison with test results it was determined that the experimental transition point always lies somewhat further downstream than the theoretical tran- sition point. The reason is that between the theoretical and the experi- mental transition points lies the region of amplification of the unstable disturbances. This amplification also can be calculated on principle according to methods similar to those previously described. (Compare Schllchting (reference 112) where this was done for the special case of the plate in longitudinal flow. ) Presumably one can obtain a still closer connection between the theoretical instability point and the experimental transition point by applying such calculations to the accelerated and retarded flow. CHAPTER XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION LAYER ACCORDING TO THE METHOD OF GEUSCHWTTZ (REFERENCE 78) a. Integration of the Differential Equation of the Turbulent Boundary Layer In order to integrate the system of equations (I8.I6), one first introduces dimensionless variables. One refers the lengths to the 9k NACA TM No. 12l8 wing chord t and the velocity to the free stream velocity U^, thus: t "" ' t S 1^: = c* = IP (22.1) Hence the system of equations (equation (I8.I6)) may "be written: — + o.oo89i^ -^ = 0.001^61 f-^'f i- dx* t3* \U J -d* djc* \ 2/ U. dx* o (22.2) First, the second equation is solved with constant values for ''"o/pu and H, namely ^ = 0.002; H = 1.5 The first approxinjation '"3 *(x*) obtained from that is then substituted in the first differential equation. From the latter one obtains a first approximation ^-,*(x*) and from that, according to equation (22.1), Ti-|(x*). With T], (x*) one determines according to figure 92 the course of H(t)) and corrects t according to equation (18.I5). Then one obtains from the second differential equation a second approximation For the solution of the differential equations one uses the isocline method which can be applied for the present case, according to Czuber, in the following manner: Both differential equations have the form: g + f (x)y = g(x) (22.3) As can be easily shown, this differential equation has the property that all line elements on a straight line x = constant radiate from one point. The coordinates of this point (= pole) are: NACA TM No. 12l8 95 i = X + -i-; X = 44 (22A) f(x)' f(x) Thus one has only to calculate a sufficient numiber of these poles and can then easily draw the integral curve. Figure 93 indicates the result of such a calculation for the profile J 015; °a ~ ^' '^^^ calculation of the laminar boundary layer for the same profile was performed in chapter XII, figure 49 ^ tahle 6. Initial values: The transition point was placed somewhat arhitrarily at the velocity maximum of the potential flow (x/t = 0.l4l). It was assumed that: Ee = -2_ = 10 V For the laminar "boundary layer was found: U t ^ = 0.141: ^ 1 -^ = 1.56 (tahle 6) Hence there results, with S*/i3 = 2.55; - = O.lU: (1) = 0.611 X 10-3 (tahle 8) The corresponding r] — value was ass'jmed to be T] = 0.1 (table 8) Calculation to the second approximation suffices. The result is compiled in table 8 and figure 93. A turbulent separation point does not exist since r\ remains below 0.8. From the variation of the shearing stress T along the wing chord the drag coefficient of the surface friction may be determined: W=2b| T dx (x= measured along chord) 96 -^kCk TM No. 1218 or c,^ = W/2 Id t § Uc w °w = (22.5) The e-valuation of the Integral gives c^ = 0.0090 t. Coimection Between the Form Parameters ti and H = tS/S* of the Tiirhiilent Boundary Layer According to Pretsch (reference 80) one may also represent analyti- cally the relation "between the form parameters t] = 1 — (u^/tr)^ and H = 5*/t3 which was found empirically by G-ruschwitz, compare figure 92. A power law is set up for the velocity distribution, of the form: H -f-L^ _ n U 1^8 ; = z (22.6) with n = 1/6, 1/1, 1/8 . . Hence results: according to the experiments so far. 5* 6 ni 1 - n\ n z dz = n + 1 (22.7) dy/5=0 Furthermore ; 1 5 0^1= ^"(^-^") s^-f)^l=l ^- dz (22.8) n _ 7,2n z^^ - z i\z = _J^ L_ = S y n+1 2n+l (n+ l)(2n + 1 z=0 NACA TM No. 12l8 97 From equations (22.7) and (22.8) follows: H = ii- = 2n + 1 or n = H - 1 (22.9) From equation (22.8) follows further i = H - 1 5 H(E + 1) (22.10) The GruBchwitz form parameter t) Is defined according to equation (l8.2) hy: 'u Tl = 1 - ■■3\2 U, With equation (22.6) t] becomes: 1 - iT (22.11) Substitution of equation (22.10) into (22.11) gives; (22.12) The connection "between H and t\ calculated according to this equation is given in the following table and is also plotted In figure 92. The curve calculated according to equation (22.12) almost coincides with the curve found empirically by Gruschwltz. 98 NACA TM No. 12l8 H T\ 1 1.1 0.270 1.2 O.i^Oi^ 1.4 0.573 1.6 0.688 1.8 0.772 2.0 0.833 2.2 0.881 2.k 0.916 2.6 0.941 2.8 0.959 3.0 0.972 Translated by Mary L. Mahler National Advisory Committee for Aeronautics WACA TM Wo. 1218 99 BIBLIOGRAPBY Tart B. — Tiirtulent Flow Chapters XIII to XX '""60. Darcy: Memoires des Savants strangers. Vol. VII, I858. 61. Reynolds, OslDorn: Phil. Trans. Eoy. Soc, I883 or Collected papers vol. II, p. 51* 62. Prandtl, L.: Ueter den Luftwlder stand von Ku^eln, Gottingen Nachrichten, p. I77, 19114-. 63. Lorenz, H. A.: Abhandlungen ub. theoretische Physik. Bd. 1, pp. k3 - hi, 1907. 6k. Boussinesq, T. V.: Mem. Pres. par div. Sav., vol XXIII, Paris, I877. Theorie des I'ecoulement tourtillant, Paris I987. 65. Eeichardt, H. : Mess^jngen turtulenter Schwankungen . Naturw. I938, p. h-Ok. 66. Prandtl, L. : Ueber die ausgebildete Turtulenz. ZAMM I925, p. I36 and Verhdl. II. intemat. Kongress f. angew. Mech. Zurich, I926. 67. Schmidt, W. : Massenaustausch in freier Atmophare und verwandte Erscheinungen. Hamburg, I925. 68. V. Karman, Th. : Mechanlsche Aehnlichkeit und Turtulenz. Nachr. Ges. Wiss. Gottingen, Math. Ehys. Elasse, p. 58, I93O. 69. Blasius, H. : Forschungsheft I3I des Tereins deutscher Ingen. . I9II. 70. Nituradse, J.: Stromung in glatten Rohren. "VDI— Forschungsheft 356, 1932. 71. Nikuradse, J. : Stromungsgesetze in rauhen Rohren VDI— Forschungsheft 361, Berlin 1933- 72. Schlichting, H. : Experimentelle XJntersuchungen zum Eauhigkeitsproblem. liig.-Arch. Bd. 7, I936, p. 1. 73. Prandtl, L.: Ueber den Re ibungswlder stand stromender Luft. Results of the Aerodynamic Test Institute, Gottingen, III. Lleferung, 1927. jh. Prandtl, L. : Zur turhulenten Stromung in Rohren und Langs Platten. Restilts of the Aerodynamic Test Institute, Gottingen, IT. Lieferimg, 1932. ■"The reference numbers used herein have "been taken directly from the original German Version. 100 NACA TM No. 12l8 75. v. Karmaxi, Th. : Verhdlg. III. Internat. Mech. Kongreas, Stockholm, 1930. 76. Prandtl, L., and Schlichting, H. : Das Widerstandsgesetz rauher Platten. Werft, Eeederei, Hafen. 1934, p. 1. 77. Tanij Hama, Mituiai: On the Permissible Eoughness in the Laminar Boundary Layer. Aeron Bes. Inst. Tokyo Eep. I5, U19, I9U0. 78. G-ruBchwitz, E. : Die turbulente Eeibungsschicht in etener Stromung bei Druckabfall und Druckanstieg. Ing.-Arch. Bd.II, p. 321, I93I. 79- Millikan, C B.: The Boundary Layer and Skin Friction for figure of EeTolution. Trans. Americ. Soc. Mech. Eng., vol. 54, No. 2, 1932. 80. Pretsch, J. : Zur theoretischen Berechnung des Profilwlderstandes. Jb. 1938 d. deutschen Luftfahrtforshung, p. I 6I. 81. Toll mi en, W. : Berechnung der turbulenten Ausbreitungsvorgange. ZAMM Bd. lY, p. 468, 1926. 82. Schlichting, H. : Ueber das ebene Windschattenproblem. Ing.-Arch. Bd. I., p. 533, 1930. 83. Swain, L. M. : Proc. Eoy. Soc. London. A. 125, 129, P- 647. 84. Betz, A. : Ein Verfahren zur direkten Ermittlung dea Profilwlderstandes. Z. f. M. Bd. 16., p. 42, 1925. 85. Jones, B. M. : The Measurement of Profile Drag by the Pitot Transverse Method. ARC Eep. I6888, I936. 86. Elias, F. : Die Warmeiibertragung einer geheizten Platte an stromende Luft. ZAMM, 1929, p. 434. 87. Eeichardt, H. : Die Warmeiibertragung in turbulenten Eeib-'jngsschlchten. ZAMM 20, 1940, p. 297. 88. Olsaon, Gran. : Geachw.— und Temperattir— Vert ei lung hinter elnem Gltter bel turbulenter Stromung. ZAMM, I936, p. 257. 89. Schultz-Grunow, F. : Neues Eeibungawiderstandsgesetz fur glatte Platten. Luftf. Forschung 1940, p. 239. 90 : W^ierstandsmeasungen an rauhen Ereiszylindern. ARC Eep. 1283, 1929. 91. Forthmann, E. : Ueber turbulente Strahlausbreitung. Ing.-Arch. Bd. V, p. 42, 1934. WACA TM No. 1218 . 101 92. Taylor, G. I.: The Transport of Vorticity and Heat Troiogh Fluids in Turtulent Motion. Proc. Boy. Soc. A, Vol. I35, p. 685, 1932. 93. Fage and Falkner: Note on Experiments on the Temperatiire and Velocity in the Wake of a Heated Cylindrical Obstacle. Proc. Roy. Soc. A. Vol.135, P- 702, 1932. 914-. Lorenz, H. : Zeitschr. F. techn. Physik 15, p. 376, 193^4-. 95. Hagen: Pogg. Ann. Bd. k6, p. U23, l839- 96. Froude: Experiments on the Svirface Friction. Brit. Ass. Eep., I872. 97- Eryden, H. L. and Kuethe, A. M. : Effect of TurTDiolence in Windtunnel Measurements. KACA Eep. Wo. 3l<-2, 1929 . 102 MCA TM No. 12l8 BIBLIOGRAPHJ ^ Origin of Turbulence (chapter XXI) 101. Reynolds, 0.: Sci. papers 2, I883. 102. Eayleigh, Lord: Papers 3, 1887. 103. LorenZj H. A.: Athandlungen u'ber thsoretische Physik. I k3, Leipzig, 1907. 104. Sommerfeld, A. : Ein Beitrag zur Hydrodynami schen Erklarung der turbulenten FlussigkeitslDewegung. Atti. d. IV" congr. int. dei Mathem. Eom I9O9. 105. V. Miees, P.: Beitrag zum Oszlllationstheorem. Helnrich Weter Festschr. I912. Derselte: Zur Turtulenz— Theorie . Jahresbericht d. deutsch.. Mathem. Ver. I9I2. 106. Hopf , L. : Der Verlauf klelner Schwingungen in einer Stromung relbender Flussigkeit. Ann. Phys. kk, p. 1, 191^; aj^i: Zur Theorie der Turhulenz. Ann. Phys., 59 p. 538, 1919. 107. Prandtl, L. : Bemerkungen liher die Entstehung der Turhulenz. Zeitschr. angew. Math. u. Mech. 1, k31, 1921. 108. Tietjens, 0.: Beitrage zum Turhulenzprohlem. Diss. Gottingen, 1922, and Zeitschr. angew. Math. u. Mech 5, 200, 1925. 109. Toll mi en, W. : Ueber die Entstehung der Turbulenz. Nachr. Ges. Wiss. Gottingen. Math. Phys. Klasse I929, p. 21. 110. Toll Tn ien, W. : Ein allgemeines Kriterium der Instabilitat laminarer G-eschwindigkeitsverteilungen. Nachr. Ges. "Wiss. Gottingen. Math. Phys, Klasse Bd. I, Nr. 5, 1935. 111. Prandtl, L.: Ueber die Entstehung der Turbulenz. Zeitschr. angew. Math. u. Mech. 11, k07 , 1931. 112. Schlichting, H. : Zur Entstehung der Turbulenz bei der Piatt enstromung. Nachr. Ges. Wiss. Gottingen. Math. Phys. Klasse, 1933 j P- I8I. 113. Schlichting, H. : Amplitudenverteilung und Energiebilanz kleiner Storungen bei der Plattenstromung . Nachr. Ges. Wiss, Gottingen. Math. Phys. Klasse 1935, P. ^8. WACA TM Wo. 1218 103 Hi*-. Schlichtingj H. : Berechnung der kritischen EeynoldsBchen Zalil elner EelTDungBBchicht in beschleunigter mid verzogerter Stromimg. Jahrlsuch 19^0 der deutschen Luftfahrt— Forschung, p. I 97- 115. Schlichtlng, H. ajid Ulrich, A.: Zur Berechnung des Umschlagea laminar /turtulent. (Preisausschrei'ben 19I1O der Lilienthal Gtesellscliaft). Lilienthal-Gesellschaft Bericht S. 10, 19i4-l. ioi+ NACA TM No. 12l8 TABLE Vn . - THE BRAG LAM OF THE 3dOOTH PLATE I '^f 0.074 Eel/5 la ^f 0.074 1700 Eel/5 «- II °f 0.k'i'> 2.58 (log Ee) o.4?5 TJ 1 Be = f „ „ ,2.58 Ee (log Ee) HI o 0.427 f 2 64 (- 0.407 + log Ee) ■ I la U Ha HI Ee=^ Of X 103 Cf X 103 cj. X 103 Cf X lo3 Of X lo3 IC? 7.40 7.13 7.63 2 X IC? 6.43 6.11 6.50 3 X IC? 5.93 5.62 5.85 4 X IC? 5.60 5.33 5.50 5 X 10? 5.37 5.06 5.23 6 X IC? 5.18 2.35 4.92 2.17 5.06 8 X lo5 4.88 2.76 4.62 2.50 4.74 10^ 4.67 2.97 4.46 2.76 4.51 2 X 10 4.07 3.22 3.96 3.11 3.95 3 xlO^ 3.74 3.17 3.67 3.10 3.68 4 xlO^ 3.54 ■ 3.11 3.50 3.07 3.50 5 XIO^ 3.38 3.04 3.40 3.06 3.33 6x10^ 3.26 2.98 3.28 3.00 3.21 8 X 10^ 3.08 2.87 3.09 2.88 3.07 lof 2.94 2.77 2.99 2.82 2.94 2 X 10'^ 2.56 2.47 2.67 2.58 2.62 5 X 10^ 2.13 2.16 2.38 2.35 2.26 108 1.85 1.83 2.14 2.12 2.03 2X 10^ 1-93 1.92 1.87 5 xio^ 1.70 1.70 1.61 109 1.56 1.56 1.47 2 X 10^ 1.43 1.43 1.33 5 X 10^ 1.30 1.30 1.19 10^° 1.20 1.20 1.10 NACA TM No. 12l8 105 o EH O Hi fa o g I O § ON J- o o § o m o VD OJ ^ H no H CO H [^ -* in OJ l>- o NO H VD -* vo OJ in ON OO ON -i* rH CO NO ^ -* OO • • • • • C C tr- I-- \o VD in ^ -* OO OO OO (M OJ CM OJ CM 1- t> OJ ^ on ^- u\ o rH O 0\ OJ t-- OJ OJ in t— H j- OO NO in J- £t^ VD 03 o in ON in OJ On t— CO co CO NO CM -* O O H M H OJ OO OO -rf- in NO t- 00 ON CJN J- VO CO H CO ^ NO ON OJ in t^ CA ?? in o OJ in o i>- OJ in t^ ON H ■^ NO 00 ON H (^ H OJ ^ ^ in in in in NO NO NO NO NO 1^ f- •H -P • O o o o o o o o O O o o o o o cvi t~- o ^ o o NO OJ in H OO t~- in en t- H * H M3 o ro o OJ rH CO NO OO H CT\ c^ NO in in H -=^ VD t~- c^ t^ NO NO NO NO in in in in in • • O o O o o o o o o O o o o o o ^ ro ^ O H r-l H o t^ H o ^ c~- in 00 in OO NO CO o OJ H ^ M3 t^ CO H -4- t-- H NO rH t- OO CO CM NO t- ^1^ O o o H H H OJ OJ no oo -=f -=J- in in in t~- OJ -* ^ O H OO NO ON in o NO OO OJ H oo ro 1-1 o On CO f- NO NO NO in in in m ,°'^ OJ OJ OJ O OJ r-\ H M H H H rH H H •H o o o O o O O o O O o o o o o I- ^ o o o o o O o o O O o o o o o o o o o o o o o o O o o o o o H UA in t- H in CO ON o in 00 H -* in in w O H OJ m -=J- -* ^ ^ in in in NO NO NO NO H H ■H H f-l r-{ H rH H rH rH H H H ,-\ ^ o in in in in NO On OJ J^ OO in -* t- H m H CO OJ in h- ON H -=)- NO CO CJN <.) o P- H OJ -* -=*• in in in in NO NO NO NO NO t~ t— O o o o o o o o O O O O o o o OJ t— ^ Oi VO ro ON OJ in H rH ^ CM in CM * H VO o en H OJ H CO NO OO rH 0\ t-- NO in in -»_n H -* \D I^ t— f- NO NO NO NO in in in UN in 8 -P i o o O o o O O o o O o o o o o Cv-l O H iH H t- ^ in H OO H in NO O f- On OO o rH M /--< VD ^ o ro NO O -=J- ON in o in o OO -=j- <Pl-P O o O H H H OJ OJ OJ ro ^ -=h in in in CM H 8 \ y d II in • \ ^ M m CO OJ CO O o 00 H in rH t— o r-[ H Rl^ H a\ 00 rrx o o CO t- in OO OJ OO OJ on -=J- -=t in in ^ ^ -* -=J- -* ^ -* -* • -b° O ? ? ? ^ ^ ^ V V ^ f Y V V Y r-\ t~- ITN OJ OO NO in ro -=f i>- in t~- ON ® ^ M0° t-- ■ VD -* H t^ OO CT\ in H t-- j- H o\ OJ OJ OJ OJ H M O o O 0\ ON ON 00 CO CO H r-{ H H H rH r^ rH rH o o O o o o H in in O rH NO H OO H H ON J- o o J- |>- rn o CO NO ^ OO rH ON NO H NO <^ o Ml-P H i-l OJ m ro J- in NO f- t- CO ON ON ON o O o o o O O o o o O o o o o H o ON o o 106 KACA TM Wo. 1218 Laminar /////////^yy//y////////yy Figure 71.- Turbulent Laminar and turbulent velocity distribution in pipe. Velocity t/Z'^y r iJL i Time t Figure 72.- Fluctuation with time of the velocity of turbulent flow at a fixed position. WACA TM No. 1218 107 Velocity y 1 Cy ^ Re crit ***-^ Figure 73.- Drag and drag coefficient of a sphere. Subcritical Supercritical Figure 74.- Flow around a sphere; subcritical and supercritical (schematic). 108 NACA TM No. 12l8 0, '77yy777yvZ7y77:^pV7y77777777777' laminar i turbulent ^crit. Figure 75.- Laminar and turbulent boundary layer on a flat plate in longitudinal flow. y uM V \ \v' 7 '■'■' y Figure 76.- Transfer of momentum by the turbulent fluctuation velocity. y u(y,-U_ ti(y) nn i-^u Figure 77,- Explanation of the mixing length. NACA TM No. 12l8 109 to l<^ V 0) a, •t-i u, 5 o o o a ^i^ 5 a l^ g OS r— 1 o • i-H +-> ;=) ^ • rH !~f c^ •i-H ^^ t:! ^ •■-1 CJ o . — 1 ir^ 0) > ^ 1 — 1 OT ^H 0) > • (—I a Ci D c\j , on 1^ Q) u It, •I-H c^ 110 NACA TM Ho. 1218 i 1 > 1 1 \ la^^SS 1 V " g 1' ii 11 II 11 " CO ^t . -. » r 3 ® • \ •s «l \ ^ V ^ X '^\U •• V *^ § 0:1 t>- Q) ;=! ho •l-l NACA TM No. 1218 111 lA" ( 13 12 II 10 9 \ * n 8 7 S75lOCj y Smooth n? 1 O — D u/ e " - 252 6 5 4 3 m " = /^^ 1 1 \ 3 ® " = - C " = • 1 60 ;::? o o <1 smfh \ 'J' OU-O 15.0 I ^ \ 7 vJ ) >/^ ^^< c N: i 1 1 ^T^ i^ A 1 1 r ^ ^S ^^^=0- 0.1 02 03 O.A 05 06 OJ 0.8 Q9 10 Figure 80.- Universal velocity distribution law for smooth and rough pipes. 112 NACA TM No. I2l8 ^ lf5 > IfJ CV4 ^ <:i 0) i^ ^ cc> o MJ g CO <0 0) V 5 «)H o M- ^* § r-H ^1 O >• ai -^ CO ^ CO ^ 0) u f— 1 §? to 0) > Co •a OS IJ > • OS CX) <u Jh Jii •-^^ •1-1 P-H c^ cs • CX) V ^J WACA TM Wo. 1218 113 0) a iX bO O U £ '■h-i o . — I CD O +-> w w C\l 00 0) tso 11J+ NACA TM Wo. 1218 T3 ft 1— 1 0) > >? 3 ^ s \ T ft ft \ rough flow. \ 7 \ \ aw of i •oughn< \ -fc ^ ,—1 M 0) \ -n <, fc^ \ <a c 1 CO CD f^ .i4 o •rH -t-> O w m o o to w ho O ■ rH iZr^^^^'^^^^/.T/r-^^ = ^°'Soigis- ^ = / KACA TM Wo. 1218 115 R ^^^ ^->^%%^^ S2 / / / /7V7777777. 4 (a) Roughness tunnel. soo ' *00 350 300 250 0,1 0,3 Ci5 ffl 2,0 1,0 0,5 0,3 0,1 (b) Velocity distribution in the section A A. Figure 85.- Measurement of the drag of an arbitrary roughness. / / ^ I Sao ot / 1 \ / ^ \ KO ugl 1 f^ 1 ^ \ h. ■ »H \ \ ft O o ■ 1 1 \ \ o Hi Figure 86,- Calculation of the turbulent plate drag. 116 MCA TM No. 1218 c^ iS «M~ ♦Jn ^ <»: !^ Q "Si * K tv <fe 5 cy 1 vi. 1 II J^ V. • • N ^ • $) .?* K Ur ^ 1 <o «a •0 5> N' «M' ^ ^ ^ 12 ^ ^ •iH <^ <5rl <5:, II II ^'^ »> • • cs *! ^ N 1 I 1 ^ ^ t^ <5> -5» ca <^i »4: *==>1 ^ Qc ;S5 • « H ^ / // // h 7/ // 1 .^ // // // // 1 4 1 —t — -/ — -/' ■/ k, /// /'/ / §1 I' i/l -fU— -Pih^ / M \ 1 I 1 1 / — 7 / .' ^ — ^ 1 — ^ k — / C- 1 • ^: / 1 / y 1 1 k^ / / - 1 5 / y / - -L r / Ok « & ^l'^ •n *M •^s (o to A Q: o a o I— I +-> f— t a St <M O O <D cd i-H bD a) u X) 00 CD U bO •f-H NACA TM no. 1218 117 II 0) ^ u o cu 8 S cd u I V "• ■a lo S >^ ^ o r 1 10 C f "^ n t S men M^ ■§-!? / S S ;5 S^ ^ / m j-4 tl t> !-i .rf / Qi tc! 0- <! -o A^ -t-> A A / 0) If ^ f W . / •a s Ci -t-> M W Lfg. 1 ;rei, Hoi ook 1925 364 264 / / ^- w XI / / 0^ So ^^^ / ■pi ,,Ci >H a -^ CO S / m (Uii h-f)HOf-<^c; / 10 U hot! OT > -2 "^ S ° -3 / < / ti .^ m n, ■^. <!' ai ^ ^ ^ / i /^ 0) cu I T r^ f< 4 § f k-A K ^ 1 llO **-• 1 i I s 00 u • f i 10 J f if t( / yj / J / cc! ^*/'/ 0. g -S ' V ' W^V ^ -3 1 + 1 7 7 ^ i < + *'q '^ L '" I ^ 3 r^ H V Kr i -t ? ► "l ^ ■^ fcf jt > r • k- \ 1 hD 1 a , i V ^ \ y^ ' / ( X V ^ t 1 ^^ y - J F 13- 3- - M =1 < / ^*'^'^*i*^ / 9 :?" / h % « .^ ^ l> / ^ in \ \ / / » -J < ^ ; ^ / /" 0) ra cti s o ■r-l fH CCi O, a o o o o .—I hD oi t-l xs Si CX3 CO hD no 10 "1 <\J 118 MCA TM No. 1218 «o ml O o d o o o in * "5 If) N Cm «S o' O" C5 d — 0^ o NACA TM Wo. 1218 119 Cn — *■ 11 Grain s Izes 10* S S to' a- yd Figure 90.- Drag of the circular cylinder for various relative roughnesses. Figure 91.- Velocity profiles in the turbulent friction layer with pressure decrease and pressure increase (according to Gruschwitz [78 j). 120 WACA TM Wo. 1218 -->^^ X M K) -*- m \ \ m CD •C = , . w w CD = • - Eh + X • \ \ ^ \ \ i • \ cvj cvj i^ 03 03 ^s. « c5 05 c:5 C>5 05 f> ^ i 1 X + X II c o a* <D •a u o o o < u 0) +J 0) u ci a B u o <D 1-1 — ^ Oi — 1 s^ Oi I • r-t -t-> ni u CD (XI Oi Oi u O II WACA TM No. 1218 121 01 OA 0.6 0.6 W Laminar separaiion point 0.2 0.k Yd W~1.0 Figure 93.- Result of the calculation of the turbulent friction layer according to Gruschwitz, Example profile J 015; Ca_ = 0. 122 NACA TM No. 12l8 Figure 94.- Free turbulence: free jet and wake. Figure 95.- Plane wake flow; explanatory sketch. NACA TM No. 12l8 123 >o < ■0- -0 V «o \ \, c> ^Is.* N \ c: H i all -• c \ .c> if^ y J V S^" / X Q 1 - x < • a o ci 0) a 0) a o <n •i-i cii o o sT o 0) CO Oi v bO 12 J+ NACA TM No. 12l8 n^fii *-x ^-?2 Figure 97,- The free jet boundary; explanatory sketch. / u / / / / 1 i 1.0 0.& as OA o.z -2 -/ n * i ^ = r = ^ Figure 98.- Free jet boundary; distribution of the longitudinal and transverse velocity. NACA TM No. 12l8 125 0:^?^ o .1-1 -t-> cd -3 o .—I Qi o a m o n o w U o o 0) «(-l i r-l a (U 03 as 0) 126 KACA TM no. 1218 ^o;Po,g^ ^'■^i.Pi.Qo ^ ■■^z^Pz^92 Figure 100,- Determination of the profile drag according to Betz, Po^^^o ^■P2,92 I'Pi = Po Figure 101.- Determination of the profile drag according to Jones. RACA TM Re. 12l8 127 ■^^o v/y/. Figure 102,- The stability investigation of the Couette flow. OLS U[^) Given Stable J c/<Z? Neutral ,• c; -0 UmS ■crit Figure 103.- The neutral stability curve as result of a stability- investigation (schematic). U(y) 1 ^m -3 //?////////// ^y UMi Um . _, :J » 7777— yT. Figure 104,- Basic flow without and with inflection point. 128 NACA TM Wo. 1218 Staole Profile with inflection point ^ /Profile without inflection point ^®crit. ^^crit. •- /fe= Um^ Figure 105.- Neutral stability curves for velocity profiles without and with inflection point (schematic). P/q D = Pressure side 5 = Suction side (J = Transition point Figure 106.- Wing profile with pressure distribution. WACA TM No. 1218 129 Figure 107.- Approximation of a velocity- profile by a polygon. ]fi7^dy-0.mUm<^ X = /.356 = 893 7f!r I 1 1 I 1 1 I 1 1 1 1 1 'rt-r Figure 108.- Streamline pattern and velocity distribution of the plate boundary layer for neutrally stable disturbance U(y) - basic flow; u(y) = U(y) + u'(x,y,t) = disturbed velocity distribution; A = 27r/a = wave length of the disturbance. 130. NACA TM No. 12l8 fW<fy'iitW(i, £ ' Dt ^¥7 .^^^-0,itl9 1. Transfer of aaln to secondary ■otlon 2. Diss Ipatlon Von Haupt • ore «ent as 1J0 Basic flow TTTT^fT// In n n -I S)^Mt2 tttPu; iJn /' -us Q.ZS «f -^ -2 O 2 -^ Disturbance amplitude Energy balance Parameters of the disturbance: V/ave length x = 13.5 6; a6 = 0.466 Velocity of wave propagation c = 0.35 U m Angular frequency /3 277 T 0.163 m UmS" 893 Figure 109.- Neutral disturbance for the friction layer on the flat plate. Amplitude of the disturbance velocity: u'(x,y,t) = u-^(y) cos (x - P^t) - U2(y) sin (x - p^t). WACA TM No. 1218 131 /;/'^if dhL Urn dy' 0.5 1.0 ulUm S iO* Z I 5 1 10^ Figure 110.- Neutral stability curve for the friction layer on the flat plate in longitudinal flow. 132 NACA TM No. 12l8 1 // /// 1 // j / / 1 f i 11 ' 1 0) ■ . — 1 +-> 1 • — I /// / «^ /// w /// 11 P /// //// // / / / Ly^ /f / / T/" ^jtz ss / / 1 / iX v.",>^ / g ' / i n 09n 1 '-v ,/ ?" / /J // ^ 1 V/ II ^^'^ J/ J 7 ^ / yvl Qoa ■ 1 , / / Si \y/ / "'teJr \ ^ y/-^ V^7 ^^P n ^ L v ,^^ , // L / / ^^ £ ( V ^ ^^^ y ^ IZ^^ i^T" y*- /?/ «o Cb ■«5: C! f^ * •rH ^<^ *^ 1^ ^ s ^ o il ft «N u (U >> V. «3 1 ^ . C>- 05 ^ ^ o C 1 — 1 •O IJ m 2 T3 - -^ § ^ cu V ^ 5;^ -4-> Oi «Nl U (U O Ph ^M •♦ W^ Ci <i^ a >^ > ^1 t3 ts. iz) CD *o fO X! o 05 O 1:^ o5 <N W 1 — 1 cti ►fi £::> Hi »* CD c^ S >o 1 T-H 1— 1 ^o T-H CD u «\J d bD cvj fc Car WACA TM No. 1218 133 / // f^' W-s / /". . 'zrit 7 — 5 / / / ? 1 / in^ - / cl 1 ol dl col lU / 7 / / / ^ y / -10^. ^Xp Figure 112.- -8 -5-4-2 Z it 6 The critical Reynolds number (Uj^^s A )^^^|. as a function of A P4- 13^ NACA TM Wo. 1218 Figure 113.- Stability calculation for the elliptic cylinder of axis ratio a^A^i = 4, If). 1.0 Crit. 0.8 0.6 O.h 02 \ -B ^ojzj. , \ Flat plate \ ^"-^ i^ \ — - ^. \ — _ • — \' V ^^, s V — ^ ** ._ — — #2 5 ^o^Z 5 l[f Z 5 1QTZ 5 iO^ Figure 114.- Result of the stability calculation for the elliptic cylinder of axis ratio a-i /b-]^ = 4. WACA TM No. 1218 135 u %-' 7 10^ 10^ 10^ Uo Uo Vi 1Q^ 10^ Re= 1(^10' ^- Uo Uo 8 Flat plate Uo-t. A = laminar separation point (independent of Rg - -^) M = maximum velocity of the potential flow. Figure 115.- The position of the instability point as a function of the Reynolds number for the elliptic cylinders of axis ratios a;^/b^ = 1, 2,4,8. 136 NACA TM No. 12l8 Symmetrical Joukowsky profile: - = 0.15 Uo h 1 - f „ L)/f«!J2L 10' R ^^^yHiT ^— '^N,,,,Af '° «^w^~;r" ' ib^ H Ca=0 Uo =0,25 .inB^o" t R 10^ !£i Uo Uo Uo Flat plate Uo Re-<lf Wf 2-10^ — t— U„t, A = laminar separation point (independent of Re=-^ ) M = maximum velocity of the potential flow S = stagnation point Figure 116.- The position of the instability point as a function of the Reynolds number for a Joukowsky profile for lift coefficients of Ca = to c^ = 1. NACA-Lanelev - 9-15-49 - 350 OO J MD 1 oo H H H H H H +3 oJ -P H S ■ EH ■H CQ (D H P4 • 1 •H CO 03 !V ^ m v ;4 03 \ 03 fH \ 03 U V < a <\5 03 l> 03 u »\ « o ^) K < r g •H -P o 03 o 1/ 1 -p o CO ^ § -p o o g' CO ta n ■H o _pq ^ CO CO -P tn •H H m :s tn •H H 03 (D o -P CM ^ ® o -P OJ ^ ^ j:! H <_ 1 ^ H _^ •H H O ■H H O •H ^M (i< •H ■ u U P^ •H • H CQ -P 1^ O O OS CO -P H O O CTs ON ® (D CO S H i-:i ® 03 CO S H -p (D ^■^ , ^H ^■^ ^H •-3 +3 ^ M < -H +3 ,Q W < ^ •V (D 3 t>» < P< t>s ^^ g i-:i &H pq a < o iA EH m S "< H h rt cu 1 CM ( on H -^ H H H H -P • -P 'I EH 5 •H H EH U •H CO 03 03 K O U V. 03 U \ . §» (D \ ^ > ® 03 i< J f? d o l<< ^ Pi / "g o +3 3 7 1 o -p / p o d ;3 o 1/ O 00 >? / O KI ^ CO U ■P o JP g" CO ^ 03 •H H OQ ^ CD tH H 03 m > ■P CVJ rQ <D 03 > -P CM rO (D O ^ H < < O ^ H _< -p •H H O •H H o PI ^ P^ •H • •^ fH P^ •H ■ © O d O CT\ ^H 03 d O ON ■^ ra -p S-4- (D CO -P S-d- cl O ^ o^ I^H 1=1 o _ ^ .Q o ® w S H m 03 <D CO S H ^ ^-^ • ^H a ^■^ , ^H ^ -P rC" w < -H ^^ -P ^ M < -H •S t>5 ^ ^ Eh <D 3 !>s 9 h g 1^ EH m fe <; •P Kl EH pq s < H '03 Fq rt H H •H -P •H -P Cd cd • (d cd • -p H t>s id © -P iH >5 Td © © tH © ^H © Xi l+H © liH © fn © ,C5 <»H -d ^ O -P ^ OO © ?H Cd ^ HJ -d ^ -P ^ (d © ^-1 cd ^ 43 CI © ^ o ^ •a Cm -P d © xj Sh ■H Cm -P •H >■ ca -p tH © d •H > CO -P Cm © d d O -P H •H ■H -P H •H "d © j:^ -d © i i>j . 'd © j:! Td © n >a . (D lb -^ -p o o H •p ft-d © "d ""^ hj f^ 43 Pi-d -p d w^ V 03 S Td q -p (d © -p ■H © H © -d -P a Kixl ■5J (d TJ d -P 03 © 43 •H © H © -d © c © © ^^ m 3 H a rt © . •H ^ ^ © d © © ^ CO 3 H a d © ■ •H ^ :d ■d d ,Q -P H •d d ,a +3 H -P © -H © cd _ -P © -H © (d ft CD M N o -H © H -p n u > -p a d Pi CO m N CD -H © d -P CD u > +3 a d -H CD -H -H m -H •" ^-ST^-^fe tH to u tH P4^ tH -H ^ Cd X > © Cm ta U © Cm © ^ H P^-P CO H CD © ^ ^ en :* ^ * ^ ^ t>» © ^ CD -p o a ^ -p -d 1-) © cd +3 «j a od^ H 3^43 EH § ^ +3 -d H © 3 (d tH O Cl H P! Cm Ph Cd Cm d H 3 Cm Pk Ih © ,Q CO © "d 1-3 m -d © CD H -H >s © -p CD © ,a cd © -d 1-3 CD CO H -H >> © ,0 §g ^ 1> -^ ,C1 3 u U > " < ;i o 3 PL, +3 3 © CD © -d d <; ^ g^^5 § © g © -d d ^ H EH O N CO ^ ,C1 rH EH N m ^ h «H © O ,0 3 _ ^ -P +5 -P •< -H 3 ^ -p +3 43 ->-H © © U -P © © ;h 43 40 rt • -P 4J Ch > © © cd 40 d • +3 -p Cm > © © cd Ti -da ^H © Cm © © • 5^^ Td -d d <iH © ^ © © • ^^^ (d -p -p H 'D -PC 3 > H -H Cd 4J 4J r-j CO -p d 3 Is H -H ^ d ^ d -P eO © m t> ^ CD >, CD 43 (d © CD t> ^ CO t>, © '^ m ti U r-{ © © 03 13 P <+H -d ;h Ch fH S Td CO d © H H © © S Td 3 <M Td U Cm u 3 "-3 © f^ © +3 ^ © -d rj '^ © ^ © 4^ j:^ © -d H ^ -H Pi a m Td -P to ,ci H Pi a CO -d 43 to 43 d r-i p! ^ tH K) © tD cd © t>j H "d © cd pS ^ -H «) © CD cd © >^ d-d © cd Ph 3 a: ^^ Cd a ^^ ed g © -H cd -P CD ^ ^ a © -H cd 4= cd ^ ^ a © © ^ (d -d -H (d +3 ^ © < ^ X! fld -d -H id 43 ^ © U Cm tn > m © -p H © cd © -P 3 ^ EH H Eh t> © © 43 rH © cd © 43 3 ;h EH H < a ■H d tH 4^ (d © Cm •d > H-5 cd © <(H •d * © cd © CO H ti ^ H h (d 2 9h p,-P < a £ © © cd © CD H ''^ ,d H ^ cd d EH P<43 < (d E © ^ H >• ^ 3 -H H t» H H •H +:> ■H 4^ Cd (d o3 cd • -P H !>5 Td © -P H >5 -d 2 © tin © ^ © ^ <iH © I^H © Jm © jd <»H -d ^ 4J ^H Id © ?H cd ^ HJ Td jj 4J f^ cd © fn (d §> 43 C! © ^ fH ■H Cm -P d © ,c fH ■3 CM 43 •H f> DO -p (kH © d d nH t> CD 43 <(H © d -P H T- •H 4^ H ■H Tb © ^ "d © P >~, . "d © ^ -d © g t>s . -d "^ +3 ^ -P Pi-d © -d "-3 4J ^ 43 Pi-d sJ 3 -d ri -P <d © -P •H © H © -d 43 d wxi cd cd "d d 43 cd © 43 •H © H © -d © d © © ^ ro 3 H S fl © • ■H ;d Id © y © © ^ Vi CO 3 H a d © • ■H ^ P -d d ,Q -P H -d d ,a 43 H -P © -H © cd 43 © -H © cd Pi tn M N m -H © q +3 ID u > -p g d P4 CO »] ^g CD -H © d 4^ © u >■ 43 a d -H CD -H -H CD a ^ •H P4^ -H -H ^ cd H > © <*H t)D (D 'i-i ■^ '^^T^'S fe <tH to ©Go > ^ u ^ ^ >> W H <d H P -P m H CD © ^ CO +3 H P HJ CD H CD © A © -p §^ -p Td H © cd s ^ 43 nd H 'H Ph o Ch d H r! Cm Ph Cm d H P cd ent thr bule andt e Gr dary t-S CD CO Cd ent thr bule andt e Gr dary t-3 m 00m ■p -d © H -H t»j © 15 l>j © CD gl CD d > 3 ^ ;h (> •^ ^ ^g^&5 § © CD © -d d ^ -^ g^ll5 § © m © -d g -ci H tn N CD ^ ^ H EH " N CO ^ f-( tin © ,Q 3 ^ -P -P -P •v -H ^j Cm © ^ 3 ^ 4J 43 43 -ntH © © ^ -P © © fH 43 -P d • -P -P ^^ © © Cd 4J d • -p -p «-i^ © © Od Td Td d ■^^ ^ r-\ -d nd d 5S»^ ^ © Vi © © • Cm ts <iH © Cm © © . hh ;» <d +3 -P H tD -p d 3 > H -H (d 4J 4^ r-H oa 43 d 3 u ^ r-\ -^ rt d a d +3 cd © m t> ^ CD 43 (d © m i> ,Q CD ;h 3 -d m d © ?j H © © cd "d P =H -d U <M -d CO d © ?-i H © © CD "d 3 Cm ■d ^ Cm •^ © Jh © -P ^ © -d rj ■^ © Sh © 43 XI © "d I— ,9 H P< a ra Id -P to ,^2 H Ph a m "d 43 60 43 d H ;i ^ -H «) © 00 cd © >s H "d © 03 Ph § « d ,ci ^ M © CO cd © (>, d-d © to P( 3 s © cd E © cd E © -H cd +3 cd H h ,Q d -H ^ © -H ed 43 cd ^ ^ a ^ (d -d •H cd +3 ,c: © < 3 ,a od -d -H cd 43 ^ © U Cm < a Eh t» ra © -p ,-1 © cd © +3 3 ^ EH H Eh > CO © 43 r-l © 08 © 43 3 V( B^ H •H d •H +3 od © ftn •d > 43 cd © Cm f © 3 -H © 03 © E H 'd J^ r-{ U 5) d Eh P<-P •< (d g © © Od © CD r-| -d ^ H ^1 3 d ^ 3 -H ^ Pi H > Eh p<4j < cd H > CX) H OJ H O EH c:? f- ■ 1 o 1 CJ H • M o H OJ H ^ H H • 1 H -P • © -d o •H £Q O •H © CO O © CO s © ^ © 03 \ ^ u ?H Fh \ ® © p> 1 L <1^ © .<\ ^ s o < \S % k5^ i-^l M U JVJ w 9 1 g © < g / ^ +3 ^ +3 i § g !/ o 03 CD o Fh © o U W 1^ . CD 60 +3 -p m 60 -P a •H CO H m ^ ■H 00 H CQ ts !D > -P OJ < m ;* -p OJ <; © O fl H 1 © o ^ H © •H H O •H H o -p !h Ph •H • !h ?H pq •H • o © H O ON © © :3 O ON H CO -p ^ ISI-4- 1 CO -P a^ & fi! O C^N fj o On a © © CO S H © © CO S H o o ^'^ • ^ H ^-^ • H -p ^ M <; -H ■p ^ w < -H 1 o U o ^ o u O h (D 3l ^ g^ fd 3l ^ ^ 3 s' o ^ m !^- 1 H 1 H oJ H H 1 m • ri O •d o © •H •H -P CQ O <D (D S U m CO \ <D f^ ^ <\^ © 4h 1 m -P / 1 -P c o o a © 1^ . (30 %■ tH +3 00 ■*: ^ CD > H CVl a © o ^ H s U •H H o © fH N •H • a CO -p :3 o O ON S © © CO S H ra ^^ • H Td -P ^ 1x1 < ^ H O fn O U o ^^ ^ w H H •H -P •H -P ed aJ • 03 53 • -P H >i ti 2 , +3 H ^ -d © <D ^t-i <D U (D ,d vi © <« <D h © Xl <*H id V| o -p f^ (d © ?H <d ^ -P o -d ^ 3 © h -p 53 §= +3 O do ^ o ^H ■a 0_| 4J d © X5 O u •H Cm -P •H >• m -p im (D o d •H > 03 -p !Vh © d o d _, o -p H iH o -p H •H "O (D Jh 'O O Q >j o . •d © x! -d © c t>5 O • © tJ T-343 O O ^ -P ft -d © -d ■^ -P o O fl -p ft-d 4J S «)^ O 03 S -d ri -P CB © -p •H © H © -d ^ § ^^5 o 03 © -p •H © H © -d © d © © _S • •H ^ ;3 © g^ i © • ■H x! ;:! ;h m 3 H a d -d d ,a -P H ;h 03 d -d d XI -P H ■P © -H © OS o -p © •H © ffS o m -H © q -p m Jh > -p g d ft CO tjD N © d -p u > -p a d o •H 03 O -H n: •H m o •H 03 O -H ^ H > © Ch tiD Ih iH ft ^■^^ ^H Ch ^ u © Ch O © © Hh O > ^ > H ^ l>5 E © ,d 03 Is XI tE -H XJ +3 m 5- 03 © xl m -p o w g o 03 H P -P CO H © d ^ +3 -d H © 03 -P o o H sp s ^ +3 -d H © 03 03 Ch O d H ;:i o l+H ft 53 C|H o d H 3 O Ch ft -P m ent thr bule andt e Gr dary -d 03 o o 0) H -H l>s © -p 03 -P d -p © -P Jh '3d ? •-3 03 O O 03 H -H >s © rQ § t> U > '• XI © XI 3 © r^ 1 > ^ > •> < ^g^^59 ^ O © ID © -d d << ^ g ^^5 1 ^ o © m © Ti d ,^2 H t^ o N CO XI o XI H EH o IS) m XI O U '+-t © O ,Q 3 XI +J +3 -P •> -H ^ tH © o X" -P -P ---H © © Jh 4J XI -P © © t< -P -P rt • -p _, +3 m ^ > © © 53 ■p d • -p +3 W -S > © © 53 O rd -d d tH © l+H © © • =H 5§^:1 Ch o -d Td © <^ © d © tw +3 53 H o 03 -p o -p H tn +3 d o ?! > o H -H O 03 -P O +3 ^ m O H -H g d o -P d o > g do -P 53 © ta t> rO o ro o O t>j 03 -P 5! © 03 >■ rQ o 03 o O t>, 03 o -d in d © ^ H © © 03 fd 3 <+H •d o a h s O © ■d 03 d <D © S -d ^ H -d o ;.^° ■o © ;^ © -p ^ © -d rj ^-3 © u © -P XI © -d H ,Q H Pi a CD -d -p CO XI H ft a 03 -d -p 6D d rQ tH K) © CD 03 © >s d "^ 2 03 J3 XI •H g' © © 03 a ^ 03 o a m 03 03 >i _ ■d ^ ^ 03 o a H ^ ,Q d -H C © th 03 o -P d © © H ^-"g •H +3 ^ © x5 x" a ^03 -d -H 03 -P ^ © < ^ o XI 03 -d -H 03 +3 xl © ^ Ch EH t> ro © +3 H © 03 © -P O 3 ^ EH H tn > 03 © -p '^ © B-i H <; a o ■H d ffi © +3 o U •H -P 03 © o <4H •d > -P 03 © o iV-i •d > © o3 © ra H "^ ^ r-\ h S3 S O E <D © aJ © m ^ -d O 1 s u S -H XI H fH ^ O tH Eh ft -p < o o3 ft H S E-> P<+3 < o H t» • H •H S3 -P d -P 03 © 2 o 3 © ^4 XI o -d © -P 03 U •a © X3 "in -P O <iH -P •H i> 03 -P <iH © o d O -p H T •H -d © X) -d © F >5 O . © "5 "-s-p o O ^ -p ft -d -p § -d d +3 o © •H © 03 03 -P H © -d © d © © 3 H a © • •H x! 3 U 03 d -d d XI +3 H -P © ft 03 M) M © R -P •H U © ai o -p g d m ■H 03 o •H 03 O -H ■H ft ^g > U ^H tlO u © © <+H O ■P H t P +3 03 H 03 © xl 03 -p -d H © o I^H ft O ■hh O d H :i o 03 -P © -P Sh & t-3 03 O O CD ■P t XI 03 © ■d ^ H -H f>i> CD 03 © r^ 03 ^H (> -v ^ '^ §^||5 1 § O © CD © -d d X" H Eh o N 03 xl o ^ Vi © O X! -P x> -P © © -P --H M -p -P d • -P -p PQ ^ © © 53 o -d -d d XJ Ojr-I +3 03 r-l "in © tin © © • Vi * O 03 +3 O -P ^ CQ o _ H -H •p d o *j ^ d o +:i 0^ © 03 t> X" o 03 o Jh d © -d 03 d © © 03 "d ^ H liH •d o ^ '-^ © U © +3 X! © rrt ^ ,Q H ft a CD ■d -p 60 -p d 1-1 ?i XI ■H §> © © 03 P< P 03 03 © >s -d ^ ^ 53 O g H 03 O •H u XI x> s © •H -P © o X5 03 -d -H 53 -p X) © < a o Eh t> 03 © +3 '^ © Eh H 03 © -P o ;h •H •P 53 © O Ch ■d !* © 5! © CD '^ -d O B © XI H ^K § ^ S -H EH Pf -p < o H S> (X) H O EH < M •H O o I N -p •H O •H O CQ CO H OJ H O ON <3; -H -d •H <D m © s -p o 05 U -P 04 H H O (D I O M o CO w ^s ^ CO H OJ O On 12; -=1- „ ON •a; -H O U ^ < •H CQ <D m W s ■p o ■P m H H •H -P •H P otf (d 03 Cd • P H (>s -d © P H t>s -d ® ID tH (D ?H ® ^ Ch © "M © ^H © ,d iM -d ^j o p ^ CD ® f^ ed ^ P O Td fj P ;-i cd © ^ cd ^ p P! !D ^ O ^ •a Ch P d © ^ fH •a Ch P •H > CQ P <iH ® o d •H t> ro p Ch © d OP H 1 •H Op H •d •H -d (D X) "d ® !>s O . ■d © ,d "d © i t>j . © r,d »-3p) o o P ft-d © -d "^ p ■H P ft-d p fl «)^ o 03 3 -d q p ed © -p •H © H © -d P d ¥^ <d 3 -d d P (d ® p •H ® H © -d <D R O S _. ^H CQ 3 H a rt © • •H ^ pi © d <D <D Jh ro 3 H a d © • ■H ^ HS -d d ,5 P H -d d ,n p H -P ID -H © od _ P © -H © cd CO -H ® R p ra U l> p g d ft ro tt) N ro -H © d p ro ;^ > pad O -H ro -H -H ro -H ■^ ^-S-q-^lfe Ch tfi U -H ft^ -H ^ ^ od n > © Ch CO U © Ch © Ch H P^p m H ro © A . ro S ^ * ^ 5 t>. ro ® xi m -p o §& p -d H p M a otf CM d rH 3 a& p -d H ® 3 05 tH o ri H li o <H ft (d <H ft -P CD ent thr "bule andt e Gr dary 1-3 n -d © ro >5 © p ro ent thr hule andt e Gr dary 1-5 ro ro H -H >> ® ,0 d ^ fH !> -v ,j=i a > 3 ^ U t> ^ < ni o 3 Pn p 3 a ^ © ro © -d d < ^ g^^5 § ® m ® -d d ,^3 H Eh O M m ^ ,£1 r-l EH N ro ^ U '^-^ ® O rQ 3 _^ ^ P> p © © P »^ -H U P ^ Ch © ,0 3 ^ P p ® ® P •»'H fn P -p rt • P P Ch > © © Cd +3 d • p p © © cd o rd -d rt Ch (D tH ® id ■ ^S'^ -d -d d Hh © <>h © © • ^^^ o (d p o p H n3 p d o 3 > O H -H 03 p p r-| tn p d 3 > H -H a d d p od o CD t> ^ o ro o t>, ro p cd © ro t> ,ci ro >, m o -d ro d ® fj H ® ® cd 'd 3 Vi -d U O Ch ^ ^ Td m d © ti i-l © © CD "d 3 Ch -d U Hh U 3 ■^ ® ;^ ® ■ P ^ ■nj © ^ © p ^ © -d i-j ,o H ft a CQ -d P «) P H ft a ro -d P 60 CD cd ® >» K "d © oa J3 ^ -H «) © ro cd © >-s d-d © cd ft 3 "ci ^^ « a ^^ cd g r-{ U fi> ti -H U ® -H «d O P CD ^ ,c a © -H 3 p 3 ^ ^ ro © © ^ ed -d -H (d p ^ © << ^ ;d (d -d -H od p ^ © U <H EH t> ro <D P H ® (d © p o 3 ^ ^ H Eh >■ m © p r-i © od © p 3 ^ Bi H <; a •H d tH p (d © o Ch •d > p cd © <M ■d > © od © CD H 'd , ^ H ^ ^ (rf d Eh Ph-p «aj o od O a ® ©03 © ro r-| "^ ,d H ^ 3d EH P4P < 3 f © U B -H ^ 8 -H ft H > H t> H H •H P •H P fld cd • Cd cd -P H (>s tJ p H t>s xJ © (D tH © ^H © ^ CM © ^ © ^H © ^ "M •d fj o P _, fn oo © !h od §^ P •d V| P ^ od © fn td S' P d © Xi o U •H Ch P d © ^ Jh •a CM P •H >• ro p i>H © d d th t> ro p) ciH d op H •H Op H •H ■d © ^ "d © s t>5 . -d © ^ "d © c >5 . © -d "^ P o o f^ P ft-d © -d "^p u ^ p ft -d p d M^ o ed cd t:) d p cd © p •H © H © -d p d M^ od 3 "d d p £d © p ■H © H © -d © d ® 2 ^^ ro d H a d _5 J, ■H ^ ^ © fi © _ fH ro 3 H a el © • •H ^ ?! -d d ,Q P H -d d ^ P H P © -H © sJ P © -H © 03 ro -H © d p ro ?H !> p" a d ft ro ti) N ro -H © q p ro ;h >■ pad O -H ro -H -H m -H " ^^-q-^fe Ch t!D ;h 4J| H P p ro H Hh tiD U © Ch © iM H 3 -P OH ro © ^ ro ro © ,d m p §^ p -d H © d g& p -d H © 3 <M ft O Ch o d H ::! o Ch ft <M d H 3 (C ent thr "bule andt e Gr dary 1-3 ro ro cd ent thr bule andt e Gr dary 1-3 ro 00m p 'd ^ H -H p ^ ^ H -H t>j © CQ §1 ro d ^ 3 U U > '^ ^ '^S^^:3§ © ro © -d d ^ -^g^lJ^B © CD ® -d d rC H EH O N ro ^ ^ r-i s^ ti ro ^ 3 ^ ^ P P P «N-H fH CM ® ,0 3 rd P p P •^■H O © ?H P ® © h P p d • p p Ch 1* © © cd p d • p p Hh > ® © fld o -d -d d ^ © n-i © © • p 5*^ 3 -d -d d ^H ® Ih © © • P cd r-^ o cd p o p H ro p d o 3 * o H -H Od p> p r-| ro p d 3 > 2 ^ H -H a d d p ed © ro t» ^ o ro o sT P cd ® m t> ,Q m ^ 3 o -d ro d ® M H © © CO "d 3 CM "5 ,'^ O <H -d ro d © M H © © id "d p "h ■d fH Vi ■^ © ^ © P _, ^ "-3 © Jh © p ^ © "d M fl H P4 a ra -d P tsD 40 d t— ,g H ft a ro -d P 60 P d r- ro cd © t>9 R "d © cd ft 3 oc ro cd ® i>5 C "d © «d ft 3 c( © 03 E © (d E H 5 ,Q d -H 5 © -H cd o -p erf H h ^ d -H ^ (D •'-i ai p 3 ^ od -d -H (d p ^ © •a; 3 rd Od -d -H oa p ^ ® ^ Ch < a Eh >• ro © p rH © cd © p o 3 ^ B^ H EH t* m ® p rH ® cd ® p 3 ;^ Eh H •H d •H P cd © o <M •d > P 5) ® Ch •d > 3 nH © cd © ro tH "d ^ H fH cd d Bi ftp <: o S O E © 3 -H © OS © m r-H fd ,d H fn 3d ft H t> EH ft p <; 3 H !> CO H H o S < UNIVERSITY OF FLORIDA 3 1262 08106 288 6 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY RO. BOX 117011 GAINESVILLE, FL 32611-7011 USA